Ideally your boss should be telling you what you're supposed to do.
But if you ask me, a process engineer's responsibility in short is to ensure that in the end the plant does what it was supposed to do from the beginning.
You're the person who knows how it all works in broad terms. You're not an expert in any field: safety, instrumentation, rotating, etc. All the experts in those domains throw their 2 cents (sometimes much more) into the project and sometimes compromise the design. That's where you come in.
These can be almost trivial issues, like you will want to have equal length of piping going to 2 parallel finfans. At some point in time the piping engineer may have to work his way around some piece of equipment that got in the way because of a change. Then if nobody pays attention one finfan sees less flow than its twin brother because one pipe is much longer. It's your job to spot and rectify those issues.
Also, make sure the plant is operable and service-able in the end. There may be an operations representative who pays attention to this but there may be not. I don't know if in 2018 anyone still makes mistakes like putting a valve 3 meters high up in the air, but I have seen newly designed reactors that cannot be unloaded as they were supposed to because of some piece of pipe running in front of the dump nozzle. Think of all the activities going around in the plant, not only during normal operation but also during turnarounds.
I wouldn't get caught in details like FATing instruments. That's not your expertise anyway and you would lose sight of the bigger picture.
During construction, pay close attention to internals: distributor plates, drawoff boxes and those things. None of the other experts knows much about those. If your plant includes distillation towers, inspect those in great detail. Spend a whole afternoon inside with a torch. It's a fun job to do if you can handle the mix of vertigo and claustrophobia. If you don't know much about tower inspection, by all means find someone who does and take him with you. It takes only one mistake (like, one downcomer with insufficient clearance -- man I spotted one at 3am at night, can't tell you how glad I was) and the whole thing won't work. You won't know before you start up and you don't want to take the plant down again for days or weeks to sort it out. Take hundreds of pictures and make sure everything is in order.
