Process Engineers have a high level of input to the production of the PFD's and the P&ID's and in establishing the what needs to be controlled.
Control Engineers develop the control strategies and details and manage (or do) the implementation.
Often the two roles are combined, that is the Process Engineer also does much of the Control engineering. For a small project that can work.
But we should also mention Instrument Engineers. They work on the measurement systems (which are essential for the control to work) and often they are they are actually the Control Engineers as well.
Mechanical engineers often have knowledge of the equipment that is necessary to be able to automate it.
Note - this site is really intended for practising engineers, If you do not know what a process engineer does you can search here or here for Control Engineers or here for Instrument Engineers.
Software Engineering, Math's and Art
This site is not really for people who think they are just Software Engineers or Mathematicians or even Artists.
But the best modelers are all of those! These guys need Admin rights on their PC's.