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Although it is not required, in nearly every case, they do. Most players are not willing to go to training camp unless they know how much they are going to earn. Teams prefer to have a signed contract beforehand, as well.
If you look at question 1.11b on this page, http://www.askthecommish.com/salarycap/faq.asp, you'll see that contracts can be signed after the start of training camp. This page, http://www.washingtonpost.com/wp-dyn/content/article/2007/07/27/AR2007072702337.html, gives an example of a player who hadn't signed a contract yet who didn't show up for the beginning of camp.
To make a long story short, in general players who show up for camp do have a contract, BUT it is not required.
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