A bitcoin address is a 20-byte hash of a 64-byte public key (plus an ID and checksum to bring it up to 25 bytes). As such, there are two theoretical possibilities with a clash:
- Two separate people own public keys which hash to the same value
- Two separate people own the same public key
Due to the way that Bitcoin transactions work, either of these situations would be enough for one party to spend the funds attached to the address.
So it's worth considering the likelihood of such a scenario. Let's assume we have 7 billion people on the planet and they all use Bitcoin. For there to be a 0.00000000001% chance of just two of the total in-use addresses clashing somewhere each person would need to have 4 trillion active addresses.
(Note that this is just calculating the chance of a hash collision; the chance of hitting the same keypair is so much more remote it's not worth bothering with).