If you set a process property in a master process that calls a common subprocess, and the process executions overlap, will there be an issue with the process properties picked up by the common subprocess?
No. The process properties are execution dependent in this case, and the common component will not be impacted.
Process properties have "execution scope", meaning that once they are set they remain available for the duration of the current process execution. This also means the properties are available across other processes initiated via the Process Call step, as is common in parent/child process designs. For example, a process property value set in a parent process can be retrieved later in a child process. The opposite is also true: a process property value set or modified in a child process can be retrieved later in the parent process after the child execution completes