On any claims made by the Brit, they have to tell them about their partner (benefit fraud if they don't) but they have to make it clear that the partner has no recourse to public funds.
Once the welfare award is made, it is up to the claiment and their partner to ensure that it is correct, to make sure there are no overpayments and that visas are not breached.
Welfare is changing now away, so what you claim now, you may not get in the future.