Basically, the legal basis of a contract is the fact that the parties agreed to be bound by it, and you can't have a valid contract unless you know who the parties to it _are_. If you have no record of who they are and no proof they consented to the thing, you haven't got much to enforce.
Copyright attatches to the work, and triggers upon the creation of copies. It "breaks closed", so if you don't have a license granting you permission to create the copy then you've infringed upon the copyright on the work. If you didn't agree to the license, your copying was an infringing action. The copyright holder doesn't have the burden of showing that the copyright applies to you, just that you're in posession of a copy of the work which by _default_ is unlicensed unless you can show otherwise.
Contract law breaks open. If you aren't a party to the contract, nobody can have standing to enforce it against you. So the burden of proving it even APPLIES to you is on the person pursuing an enforcement action.
The only way I can see to apply contract law here is to have a central distribution point that records everybody it gives copies to, and to forbid redistribution. And even then, your enforcement would be against the person who redistributed it, not against the recipient who wasn't bound by the contract. (Maybe you could get them on tortious interference (inducing breach of contract), but what if the second person hands it off to a third persion? Perhaps "receipt of stolen goods" or something? But not directly on violation of a contract they weren't a party to.)