Part of me wonders if you are getting lost in the nomenculture as I think there are several options which have been mixed up a bit in the thread.. and even as I type this I realise the names are interchangable...but anyway
1) A coach - This person probably isn't physically with you all the time but will give you a workout program to follow. You check in with them on objectives and achievements but the consensus is on you to follow their program to achieve objectives.
2) A personal trainer - To me this is someone who comes to your door and forces you to exercise.. They have knowledge around your objectives but are much more hands-on than the coach.
3) A gym trainer - This person is at the gym and will help you devise a work program based on your requirements.. But by and large it will be in a gym. They can take the form of option 1 (give you a workout and then leave you to it) or option 2 (you turn up every week and they do stuff with you). Especially with this option I think you get out what you put in; they must see a huge number of people who are keen for a few weeks and then fade away. If you can connect with them I think it would be good.
I've used options 1 (as others have said didn't find it fun) and option 3 (was a guilty party who faded away).
I always say this in threads discussing strength training but technique is everything... I was fortunate that my option 3 was a national class weight lifter and suffice to say I've not looked at squats the same since... I recently watched some cross-fit and was wincing at some of the techniques on show..there-in lies the route to injury...