Depends on the carrier, and phone. Can't speak to other ones, but I know the 4G phones on Verizon can talk & surf while in a 4G area. If you aren't in 4G markets, the Thunderbolt can do it on 3G connections (as far as I know, it's the only phone Verizon offers that can do it in a 3G area) - though the speed isn't the greatest if you're doing it.
Otherwise, a lot of the smartphones CAN talk and Surf, but ONLY if you're in a wifi area for the data-side for the internet.