Show Outlook calendar in a screen-share, but obscure private appointments

One potential workaround is to create a custom calendar view, with a filter applied. When you're screen-sharing, you can switch to this view. Here's what the filter would look like:

When the custom view is applied, Private items won't be shown in your calendar at all.

'Standard' view

'Hidden' view

You can add the Change View option to the Quick Bar too, by right-clicking on the views

This gives you a nice subtle option from the top bar, which is accessible from any tab on the ribbon

Is there an option in Outlook where I can temporarily obscure the titles of private appointments?

As asked, no.

You can mark appointments as private and then no one but you can see the content of the appointment.

But Outlook does not have a switch to make the Private Appointment disappear from the Outlook Calendar, at least not natively.