It can be just about anything. My career was spent working for a chemical company doing analysis work in the area of heat transfer. Mechanical engineering is very broad. We had ME's who did design work, who did stress analysis, who did thermal-fluids work, who did failure analysis, who did analysis in the vibration area, and who wrote analysis software for others to use. There is also project type work where ME's manage projects and utilize other engineers to do the number crunching work. Other ME's, along with EE's, did automatic control work. The field is very broad.
My days were spend at a computer terminal doing analysis work, making predictions about how long something would take to cool or heat, managing heat to control chemical reactions, doing thermal stress work, and vibration analysis. My work included writing software for others to use in the thermal-fluids area.
I was always involved with an analyis group so I never was involved in running portions of the chemical plant. Those ME's who did that certainly were.