Yep, totally not nagging, giving them room to think and then jumping in to get them talking is definitely what you want to do here.
This also isn't your day to day job so you have to keep in mind that these interviews are generally held to an hour length and you can't really give them 30 minutes for a single problem that wouldn't give you much insight into the person.
It also heavily depends on the culture of the company as well. So being able to communicate effectively and act under pressure is something that goes with pair-programming and code reviews.
To note, I have never had anyone ever leave an interview feeling bad about the company or our process, the interview is about the individual.
