This varies depending on which logical system you're working in.
If the system is both sound and complete you can move between semantic "proofs" (I prefer the term "verifications") and syntactic proofs freely (since we know there is a perfect match between syntactic and semantic validity).
When systems lack soundness or completeness (generally completeness, people are usually reluctant to put forward proof systems whose acceptable inferences don't preserve truth) you won't be able to assume this perfect match.
Some systems have both sorts of proofs in a single style of representation. For instance, in An Introduction To Non-Classical Logic, Graham Priest gives a tableaux system of proof where a single proof-tree/truth-tree is both a semantic and a syntactic proof of the proposition in question. A downside of this is that you have to accept that you'll have some proofs which branch infinitely (so, you don't have a decidable proof system). This can generally be recognized because you'll just have to keep reapplying a rule that creates a new branch, and you can see that you'll just get stuck in this sort of "infinite loop" but that none of these branches will serve to "close" the tree (i.e., derive a contradiction on that branch). It is hell on automated theorem provers, though.
So this brings me to the final answer to your question "surely a proof should have some semantic status too?". Yes, generally people think that it should. This is why soundness and completeness are desirable properties for a system to have (though there are trade-offs to be made, as evidenced by the fact that most of the more interesting mathematical systems will be incomplete). But ultimately these are meta-theoretic properties and proofs are carried out in the meta-language. I think we keep these notions (syntactic and semantic validity) separate so we can modify our system along two dimensions to allow for theories where the syntactically and semantically valid propositions are not the same.