You can log anything you want, it's your logbook.

However...employers generally want to see FAR-legal time only (and sometimes they are more stringent than that with regards to PIC).

If the guy is an MEI you can get dual-received. You could also legally do safety pilot time, but future employers would be very suspicious of SP time logged in a ME turbine aircraft...they would suspect (correctly) that the real reason for the flight was not hood time.

If the jetstream were certified for two pilots (I don't think it is), you could log SIC.

Also be aware that at interviews for entry-level jobs, they will usually talk systems about one of the airplanes you have flown...and they often pick the most complex. If you log it, you might end up trying to explain the jetstream electrical or pressurization system at a regional interview.