I favor opposed checks each round, because I think that is what the rules include, and also because it seems like a good system. It will make following a group that close for an hour quite improbable, but I think that it IS quite improbable unless the follower is extremely stealthy. The follower cant’s take 10, in my view, while the followed should not want to, so I think the DM should just keep rolling until the stalker is heard, and then proceed from there with the rest of the encounter.
If the timing is not crucial, then I agree that the DM could simply accept that someone’s check will eventually detect the stalker, and just pick a person and time for it to occur.