In some configurations of the token bridge, the bridge validators will have no ability to call executeSignatures on the Foreign side (e.g. they have zero balance on the Foreign network by intent). In this case the end user will be responsible for passing the validtors' confirmation to the Foreign side to release/mint tokens.
It is necessary to think of the work flow and the interface in the Bridge UI to provide the end user ability to call executeSignatures and implement it.