As far as I know, the directive mainly applies to officers who are sent back in time and/or given the opportunity to change established history. I don't think it would prevent someone from making an arrest in their "proper" time.

At most, it might limit their ability to interrogate the prisoner, if they can verify that the intruder is from the future and possesses knowledge that the contemporary officers can't have.

Chakotay once used the TPD as an excuse to not answer a question from Janeway.

And she just accepted it.

Now you’ve paused, thinking any action could cause a temporal paradox, or damage to the future timeline.

You can also arrest them anyway. Logically, it would fit into the timeline, since if they're from the future, they should have accounted for those circusmtances, including their arrest in their future timeline. And a scan would show a fair few forms of time travel, which you can use to verify, or any connections that you might have with other temporal entities. Time travel is a bit convenient like that, until everything goes horribly wrong.

But ultimately, nothing actually prevents someone from just out and out lying. They could easily say "oh, sorry, can't say, Starfleet Intelligence business", or "I'm here on Captain/Engineering's orders", and you'd have that delay in either case to verify the story. The Temporal Prime Directive doesn't really factor into it.