What preposition seems to fit in this sentence? What will the difference in the meaning of the sentence?
I've personally heard "your word against mine" far more frequently than I've heard "your word over mine" - I find I have an unexpected bias in that the "over" construction seems to me to indicate lesser educational attainment in the speaker.
Google ngram shows a huge distribution differential which supports my internal expectations to a large degree:
Of course, knowing the most commonly-used between the two variants doesn't answer the portion of your question pertaining to implications - for that I would say I agree with @Tushar Raj's take on the implication of the phrasing, with a caveat - the active person doing the assuming is not in fact the "taker" - the use of "over" in this phrase implies that the asker is assuming that the person to whom they've addressed the question has at some level already decided that "his" word has been deemed more credible than "mine"...
Hope this helps.