Professional

The escort won’t care and I’m sure they’ll be discreet.

Or use it for blackmail.

No worries.

The escort may not care, but your company's IT department may notice and report it to HR. Don't do personal business on a work computer!

I wonder what a public escort would be.

There's one who is currently FLOTUS.

Use your own personal smartphone for all private communications. There is no expectation or guarantee of privacy using your work computer or work smartphone if you have one. That could save you embarrassment and trouble.

I work It in a hospital, all outgoing emails are monitored for PHI. Though if I saw your email I don't think i'd report it, feels like doing so wouldn't be worth the trouble.

For the escort it is work, so it is fine.