Would I be sinful for invalidating my prayer or someone else's prayer if I felt too awkward to inform them of being defiled by an impurity because of me?

Your prayer or the prayer of another person is not invalidated by not informing them about the presence of impurity. However, you must inform them so they can purify themselves. You would be sinful if you do not, especially if you were the cause of their impurity, based on the Prophet's (peace be upon him) saying: "Religion is nasiha (sincere advice)." It is obligatory to inform someone who intends to use an impure item that is not excused, as this falls under the category of enjoining good and forbidding evil. As for what is excused (i.e., minor impurities that are overlooked), there is no obligation to inform about it, and worship is not invalidated by its use. The consideration here is the belief of the one who knows about the impurity.