For large plants, it is the Chemical Engineers who do the process design and the Mechanical Engineers pick up from there. For cookie cutter plants - say like a Budweiser brewery or a gas treatment plant, Or sewage treatment plant, the Chemical engineer does the process design of the first few plants and then Mechanical engineers do the subsequent plants. So for your case you may have worked with a company that does standard “cookie cutter plant”.
Let me illustrate with heat exchangers. The ChemE does the Process datasheet for the heat exchanger. Assuming a TEMA type, the process datasheet contains the flows on both sides for many cases (design, turndown, etc), the TEMA type of exchanger, no of passes, allowable pressure drops, material of construction, tube pitch, corrosion allowance ,etc etc
The mechanical engineer takes the above information and designs the Shell (thickness), tube sheet thickness, does stress calcs etc.
The same mechanical engineer or a different one then does the fabrication drawings.