No, ownership is to be decided by governments who use force to protect property rights. They may take advice from the parties involved in the exchange, they may decide that neither party involved in an exchange was the owner.
For instance, in most jurisdictions, a dispute between two (claimed) property owners can only be decided by them, in full agreement, with the courts stepping in only when a stalemate is reached.
But the judge can only decide the dispute, and can't, for instance, arbitrarily issue an order giving the property to a 3rd party, or decide a dispute where one of the parties haven't first come to the court seeking that resolution.
Similarly, a property owner can summon police to help enforce their property rights in the case of a trespasser, but police can't arbitrarily enforce the rights of 3rd parties without request.
Free markets exist regardless of how you choose to assign property to people. Be it violence or consensus. Ownership is to be decided by the parties involved in the exchange. 3rd partys are an easy solution.