If it's work-related, work computer only. If it's anything personal, personal computer only. This is especially true for situations where companies are using screen capturing tools that capture the screen every 'x' seconds[0].
The seeming convenience of using your work computer for everything isn't worth the risk but some people don't consider the legal implications, as you've noted in your anecdote.
Once your personal message to Bob or Jane enters an official record because it's included in the eDiscovery (even if accidental) result(s), it's over.
Better to not have a surface risk like that, than assume the process will keep your personal data safe.
0 - https://desktime.com/features/time-tracking-with-screenshots