When you finish a needle the working yarn will be hanging from the last stitch on the needle and will be on your right. The next needle will be the one that has the next stitch and will be at the right end of a needle.
I do like Jan says and use the tail to show me where the round starts. I also put a stitch marker after the first stitch of new round. I suppose you could put a different colored stitch marker after the first stitch of each needle in some kind of easily remembered order like the American flag red (first needle), white (second needle), blue (third needle) then use that order to keep you organized. Keep moving the stitch markers to keep them on the row you are on. I've never done it but it should work.