How do non-Trinitarians who do not believe Jesus is God Almighty explain how Jesus can effect atonement for all of humanity? Non-Trinitarians who do not believe Jesus is God Almighty include Unitarians, Socinians, Arians, Jehova's Witnesses, etc. This would not include, supposedly, Modalists and LDS/Mormons.

The idea of atonement in the Old Testament was that a sinner would bring an offering to atone for his sin, and the death of that animal (and thus, its life) would take the place of the sinner's. 

In his Temple, Its Ministry and Services, Alfred Edersheim wrote,


  The fundamental idea of sacrifice in the Old Testament is that of substitution, which again seems to imply everything else—atonement and redemption, vicarious punishment and forgiveness. The firstfruits go for the whole products; the firstlings for the flock; the redemption-money for that which cannot be offered; and the life of the sacrifice, which is in its blood (Lev 17:11), for the life of the sacrificer. Hence also the strict prohibition to partake of blood. Even in the ‘Korban,’ gift (Mark 7:11) or free-will offering, it is still the gift for the giver. This idea of substitution, as introduced, adopted, and sanctioned by God Himself, is expressed by the sacrificial term rendered in our version ‘atonement,’ but which really means covering, the substitute in the acceptance of God taking the place of, and so covering, as it were, the person of the offerer.


Now, this would be a 1:1 relationship, i.e. one sinner, one animal. If Jesus is just a man (or even an angel, another created being), even if he be a sinless man (just like the animal was sinless and was offered as an atonement), how can Jesus effect atonement for all of humanity (John 1:29) rather than just one person?
I would imagine that your question may have multiple answers depending on the specific variety of Unitarian that is answering it. For example, from my reading on Unitarian Universalism as compared to what I know about Jehovah's Witnesses, it is unlikely that the Unitarian Universalist perspective is similar to that of a Jehovah's Witness.

The best source I was able to find on the subject is Hosea Ballou in his "A Treatise on Atonement" available here. 

The first important detail is that a Unitarian Universalist does not recognize atonement as meaning the same thing as most Trinitarians. This is from Chapter 5:


  I have already entered my protest against the necessity of atonement,
  on the principles upon which Christians have generally believed it, by
  showing the finite nature of sin, and the error of supposing that the
  law of God required the endless misery of mankind as a penal
  requisition.
  
  Atonement signifies reconciliation, or satisfaction, which is the
  same. It is a being unreconciled to truth and justice which needs
  reconciliation; and it is a dissatisfied being which needs
  satisfaction. Therefore I raise my inquiry on the question, Is God the
  unreconciled or dissatisfied party, or is it man?


He concludes that atonement is a synonym of reconciliation. 


  To say that God loved man any less after transgression than before,
  denies his unchangeability; but to say that man was wanting in love to
  God, places him in his real character. As God was not the unreconciled
  party, no atonement was necessary for his reconciliation. Where there
  is dissatisfaction, it presupposes an injured party; and can it be
  hard to determine which was injured by sin, the Creator or the sinner?
  If God were unreconciled to man, the atonement was necessary to renew
  his love to his creature; but if man were the unreconciled, the
  atonement was necessary to renew his love to his Creator. The matter
  is now stated so plainly that no person who can read can mistake.


Assuming your question still stands: "How can Jesus affect atonement for all men?" Then his answer is essentially a statement of faith in scripture (quoted from Chapter 6): 


  I next inquire, has the Mediator power or ability, to perform the
  great work of atonement, which is the reconciliation of the world to
  God? Those scriptures, with their connections, which I have quoted to
  prove the Mediator's dependency, abundantly prove the sufficiency of
  his power to accomplish the work in which he is engaged. If all power
  in heaven and earth be committed to Christ, no doubt can be
  entertained of its sufficiency. If the whole system of law in moral
  nature be subservient to the designs of the Redeemer, and if he holds
  in his hands the power of moral government, it certainly must be at
  his option, whether men shall be reconciled to God or not.


From what I was able to gather, he does not make much of an effort to understand the mechanism of how exactly this is accomplished, but given the nature of his understanding of "atonement" I doubt that it would be necessary to question the ability of Christ to effect this change in state.

Again, I would imagine that a Jehovah's Witness would have a very different idea of how this works, presumably based on a more "traditional" understanding of the concept of atonement. However, most sources I can find on the Jehovah's Witness view of atonement are from hostile sources and are therefore not reliable descriptions of what they actually believe.