In the US this is generally governed by state law: RCW 4.24.350 in Washington state. The criminal jury is not empowered to make such a decision, but a separate civil trial for malicious prosecution would be possible. Plaintif (ex defedant) would have to prove that
the action was instituted with knowledge that the same was false, and
unfounded, malicious and without probable cause in the filing of such
action, or that the same was filed as a part of a conspiracy to misuse
judicial process by filing an action known to be false and unfounded
The basis for the lawsuit would be the objective facts that prove that the prosecution was false and malicious, and not the fact of acquittal or the subjective opinion of a juror.