No, they would say "I'm the Cop" or "I'm the Doctor". (the comparison to the role "Named Townie" is in that we'd simply care that they were the Doc or Cop, not necessarily about any results they have)
Imagine this scenario: We lynch scum today. A VT dies tonight. If the Cop and Doc decide to claim tomorrow, our lynch pool would be down to 4 from the VTs point-of-view (7 living players, 2 are our PRs, they know they themselves are town) instead of 7. Now, I don't think that's worth it. But, imagine that we then lynch a VT, and a VT dies N2. D3 dawns with 5 living players. Say our PRs claim then. Our lynch pool for the VTs goes from 4 to 2 - and the Cop has a N2 result to claim. Which, if it's a town result on one of the VTs, wins the game.
(so, you say, why won't scum counter-claim one of the PRs? Because even if they get us to lynch the real cop, they kill someone at night, we still have D4 with 3 alive, 1 scum, and the scum is obvious.)
So that's a situation where it makes sense for the Doctor and Cop to claim immediately (D3, no PRs have died). There might be others. There might not be. Just something for them to think about. The decision on when to claim or not claim should always be made in the best judgement of the PR themselves, because any advice from the town is prone to scum manipulation.