Do you need the permissions from the subjects? Specially if there is money to be made.
The short answer is "it depends", and the rest of my answer is based on US law. To be on the safe side, you'll probably want to have a release. Many contests have a motive to find the best photos possible that showcase a product or service, and the subject(s) in your photos could be inferred to be endorsing that product or service, a model release would definitely be needed.
If the usage is purely editorial, a model release may not be needed.
Of course, be sure to read the fine print of the particular contest, it may address the issue of whether or not releases are required for individuals in the photos.
Someone will also inevitably point out that you can do anything, as long as you don't get caught :)
Assuming you can publish the photo as a work of art (which is not guaranteed), then it mostly depends on the terms of the photo contest. i.e., let's put aside the question of whether or not you can publish a photo of a stranger at all, and just think about the contest.
In general, the fact you're making money/receiving prizes doesn't have a direct bearing on whether or not you need a release. Selling a photo as a work of art doesn't require the subject's permission (for emphasis, with the assumption you can publish it).
Things that might impact you in the contest terms are clauses like these:
So tip #1 is read the terms and conditions. In most contexts, promotional use puts a larger burden on the photographer than simple display, and would normally require a release. The hypothetical problem boils down to this: you enter the photo, and win. It's used to promote some aspect of the sponsor's business/operation. The person sees it and objects to this use of their image, sues. You end up on the hook for costs.
Is this a big deal in practice? Yes and no. I've never heard of it happening, but on the other hand: if terms like these aren't subject to sensible limits (such as only requiring rights for the winning photos, and/or only for use directly in connection with the contest itself) then you should probably think twice about entering the contest anyway.
Ultimately, you should check the terms and conditions for a particular contest; I would expect a contest with a theme of social documentary to be far more accepting than if you were coming up with an image to promote something or someone. Bear in mind that how a photo is manipulated also comes into play; I did hear the case where a sports centre took photos of people in the street, with permission, but then found themselves in hot water as the final image gave the impression that two of the people were a couple even when they were shot separately.