Jesus's sacrifice ("the [Christ] must give his life a ransom for many" Mt 20:28) is the basis of the effectiveness of such forgiveness outside the immediate context of sacrifice. In other words, you can't forgive if you are in hell, where you'd be if Christ hadn't ransomed you from sin and death.
Jesus' focus in His ministry is how Christians are to live assuming they have been saved, it doesn't assume their needing to be saved. How to be saved is, after all, the smallest and the quickest part of Christian life (the majority of peopel being regenerated as babies): it will do you no good if you don't remain in God's favor by obeying what the Son has been sent to teach us. In fact, according to the Bible, it is worse (Heb. 10:29) than breaking the Mosaic Law, which resulted in "death without mercy under two or three witnesses" (ibid. 10:28). "Better that someone was never born" again, as it were. Than to recieve the Adamic promise and to reject it once again, making yourself 'twice as much as child of hell.'
As for why some Christians says Jesus must have offered a blood sacrifice in order to save people from their sins, this is presumably based on Hebrews 9.
Hebrews 9:22 And almost all things, according to the law, are cleansed with blood: and without shedding of blood there is no remission.
However, this is demonstrating how everything in the Old Law pointed to Christ in some way, even down to the fact that ritual and moral cleanness came from the shedding of blood. It isn't claiming that unless there is shedding of blood, sin can't be forgiven. This is absurd on its face (God requires certain specs of DNA to leave one location and be in another, in order to continue his work of salvation), but it's also a misapplication and misreading of the verse (fails to recognize whent the author is referring to types of Christ, vs. asserting absolute requirements). Reading it by its clauses, the author is stating that "according to the law [of Moses] ... without shedding of blood there is no remission." He isn't stating an absolute requirement for God's forgiveness.
In fact, later in the same Epistle (the next chapter, in fact), he writes:
Hebrews 10:4 For it is impossible that with the blood of oxen and goats sin should be taken away.