Following in the footsteps of Christ,(16) the Christian faithful have always endeavored to help one another on the path leading to the heavenly Father through prayer, the exchange of spiritual goods and penitential expiation. The more they have been immersed in the fervor of charity, the more they have imitated Christ in his sufferings, carrying their crosses in expiation for their own sins and those of others, certain that they could help their brothers to obtain salvation from God the Father of mercies.(17) This is the very ancient dogma of the Communion of the Saints,(18) whereby the life of each individual son of God in Christ and through Christ is joined by a wonderful link to the life of all his other Christian brothers in the supernatural unity of the Mystical Body of Christ till, as it were, a single mystical person is formed.(19)
Thus is explained the "treasury of the Church"(20) which should certainly not be imagined as the sum total of material goods accumulated in the course of the centuries, but the infinite and inexhaustible value the expiation and the merits of Christ Our Lord have before God, offered as they were so that all of mankind could be set free from sin and attain communion with the Father. It is Christ the Redeemer himself in whom the satisfactions and merits of his redemption exist and find their force.(21) This treasury also includes the truly immense, unfathomable and ever pristine value before God of the prayers and good works of the Blessed Virgin Mary and all the saints, who following in the footsteps of Christ the Lord and by his grace have sanctified their lives and fulfilled the mission entrusted to them by the Father. Thus while attaining their own salvation, they have also cooperated in the salvation of their brothers in the unity of the Mystical Body.
"For all who are in Christ, having his spirit, form one Church and cleave together in him" (Eph. 4:16). Therefore the union of the wayfarers with the brethren who have gone to sleep in the peace of Christ is not in the least weakened or interrupted, but on the contrary, according to the perpetual faith of the Church, is strengthened by a communication of spiritual goods. For by reason of the fact that those in heaven are more closely united with Christ, they establish the whole Church more firmly in holiness, lend nobility to the worship which the Church offers to God here on earth and in many ways contribute to building it up evermore (1 Cor. 12: 12-27). For after they have been received into their heavenly home and are present to the Lord (2 Cor. 5:8), through him and with him and in him they do not cease to intervene with the Father for us, showing forth the merits which they have won on earth through the one Mediator between God and man, Jesus Christ (1 Tim. 2:5), by serving God in all things and filling up in their flesh those things which are lacking of the sufferings of Christ for his Body which is the Church (Col. 1:24). Thus by their brotherly interest our weakness is greatly strengthened.(22)
For this reason there certainly exists between the faithful who have already reached their heavenly home, those who are expiating their sins in purgatory and those who are still pilgrims on earth a perennial link of charity and an abundant exchange of all the goods by which, with the expiation of all the sins of the entire Mystical Body, divine justice is placated. God's mercy is thus led to forgiveness, so that sincerely repentant sinners may participate as soon as possible in the full enjoyment of the benefits of the family of God.
The concept of the communion of the saints is certainly real and important, but the scripture speaks of it in the sense that we are one body, and can pray for and encourage each other; we suffer and rejoice together. However, scripture does not say that believers can expiate their own sins or those of fellow believers by prayers or actions. Scripture, on the contrary, says Jesus Christ is the only one who atones for our sins:
The word expiate is not used in the Bible, but the definition following says it is the same as to 'atone'.
Definition of expiate from dictionary.com:
v. ex·pi·at·ed, ex·pi·at·ing, ex·pi·ates
To make amends or reparation for; atone: expiate one's sins by acts of penance.
To make amends; atone.
The scriptures tell us Who atones for our sins:
3 Iniquities prevail against me;
As for our transgressions,
You will provide atonement for them.
9 Help us, O God of our salvation,
For the glory of Your name;
And deliver us, and provide atonement for our sins,
For Your name’s sake!
60 “Nevertheless I will remember My covenant with you in the days of your youth, and I will establish an everlasting covenant with you. 61 Then you will remember your ways and be ashamed, when you receive your older and your younger sisters; for I will give them to you for daughters, but not because of My covenant with you. 62 And I will establish My covenant with you. Then you shall know that I am the LORD, 63 that you may remember and be ashamed, and never open your mouth anymore because of your shame, when I provide you an atonement for all you have done,” says the Lord GOD.’”
Some of the translations don't use the word atone or atonement in the New Testament, but the concept of 'propitiation' or 'reconciliation' is used:
that by which God is rendered propitious, i.e., by which it becomes consistent with his character and government to pardon and bless the sinner. The propitiation does not procure his love or make him loving; it only renders it consistent for him to execise his love towards sinners. In Rom. 3:25 and Heb. 9:5 (A.V., "mercy-seat") the Greek word _hilasterion_ is used. It is the word employed by the LXX. translators in Ex. 25:17 and elsewhere as the equivalent for the Hebrew _kapporeth_, which means "covering," and is used of the lid of the ark of the covenant (Ex. 25:21; 30:6). This Greek word (hilasterion) came to denote not only the mercy-seat or lid of the ark, but also propitation or reconciliation by blood. On the great day of atonement the high priest carried the blood of the sacrifice he offered for all the people within the veil and sprinkled with it the "mercy-seat," and so made propitiation. In 1 John 2:2; 4:10, Christ is called the "propitiation for our sins." Here a different Greek word is used (hilasmos). Christ is "the propitiation," because by his becoming our substitute and assuming our obligations he expiated our guilt, covered it, by the vicarious punishment which he endured. (Comp. Heb. 2:17, where the expression "make reconciliation" of the A.V. is more correctly in the R.V. "make propitiation.")
Easton's 1897 Bible Dictionary
Scripture says Jesus Christ made propitiation for the sins of the people:
Hebrews 2:17 Therefore, in all things He had to be made like His brethren, that He might be a merciful and faithful High Priest in things pertaining to God, to make propitiation for the sins of the people. 18 For in that He Himself has suffered, being tempted, He is able to aid those who are tempted.
1 John 2: 1 My little children, these things I write to you, so that you may not sin. And if anyone sins, we have an Advocate with the Father, Jesus Christ the righteous. 2 And He Himself is the propitiation for our sins, and not for ours only but also for the whole world. (This also shows that when we sin, Jesus is our mediator and advocate sitting at the right hand of God after finishing His sacrifice on the cross; His sacrifice is not being perpetually or repeatedly offered; IT IS FINISHED;Hebrews 10:12: "But when this priest had offered for all time one sacrifice for sins, he sat down at the right hand of God.")
1 John 4:10 In this is love, not that we loved God, but that He loved us and sent His Son to be the propitiation for our sins.
Scripture says that if believers sin, they are forgiven and cleansed if they confess their sins to God:
1 John 1:9 If we confess our sins, He is faithful and just to forgive us our sins and to cleanse us from all unrighteousness.
Scripture says that we receive justification and forgiveness (atonement) by faith in Jesus Christ, apart from works (including works of penance and indulgences):
Romans 3:21 But now the righteousness of God apart from the law is revealed, being witnessed by the Law and the Prophets, 22 even the righteousness of God, through faith in Jesus Christ, to all and on all who believe. For there is no difference; 23 for all have sinned and fall short of the glory of God, 24 being justified freely by His grace through the redemption that is in Christ Jesus, 25 whom God set forth as a propitiation by His blood, through faith, to demonstrate His righteousness, because in His forbearance God had passed over the sins that were previously committed, 26 to demonstrate at the present time His righteousness, that He might be just and the justifier of the one who has faith in Jesus.