Depending on your software, you can! I know Google Calendar allows you to restrict shared calendars to just time ranges, it'll show up as "not available".

Settings > Calendar Settings > Access Permissions > Make Available to Public > on the dropdown, pick "See only free/busy". Now you can share the link with your work calendar and it'll show up as indistinct blocks of time.

You can do it both ways, and you'll be able to check your availability for both work-related things and personal things in their own calendar, but contextualized.

Even my work calendar doesn't share the details of what I'm doing in a busy slot. Not particularly something anyone needs to know. If someone really needs to meet and my calendar looks blocked they can ping me to see if there's anything I can drop or move. It's not really up to them to make the decision (unless I work for them and it's urgent).

Be default, my work calendar is strictly work and my personal calendar is strictly personal. I only duplicate personal events to my work calendar if they overlap with my work hours. Everything outside of work hours goes in my personal calendar only.

Aside from the odd appointment at the dentist or doctor's office, I don't have very many appointments that need to show up in both calendars. Work hours are work, everything else is personal and none of my workplace's business.

> But you need to check both calendars when making the appointment.

If one covers strictly work hours and the personal one is strictly for outside work hours, then no need to check both. Just check the applicable one.