It would reveal your private key, which might be bound to your identity in other ways. The other person could then forge your signature on whatever they wanted.
Edit: In the case of off-chain exchange of Bitcoin wallets it would reveal the private key (naturally, since that's what you're exchanging) but for that reason you would only trade a wallet with a unique private key not used anywhere else. The only thing that key is useful for is spending the funds in that wallet. It's not linked to your identity. This is exactly like the "physical bitcoin" model (BitBills, Cascascius coins, etc.) except there is no tamper-evident hologram to prove that the private key hasn't been accessed—you're taking that on trust until you empty the wallet or hand it off to someone else.