The arrangement decides everything else, so settle it in writing before anybody starts. The two common shapes are an employee or contractor you pay, and a professional who rents a chair from you and runs their own business. In the first, the clients and the money are yours; in the second, they are theirs and you are a landlord.
Blurring the two is where these arrangements go wrong. A chair renter whose bookings come through your page and whose prices you set is not really a renter, and neither of you will agree about whose client it was when they leave. Employment law in your country has a view on this as well, and it is not one you want to discover afterwards.
Write down the parts everybody assumes are obvious: hours, what is included in a rent, who buys product, who covers insurance, whose cancellation terms apply, how holiday works, and what notice either side gives. Six lines now beats an argument later.
Decide the client question explicitly. Who they book with, whose records the notes live in, and what happens to those clients if the arrangement ends. This is the single most common thing to be left unsaid, and it is the most expensive.
On the platform, give them their own account and their own calendar rather than sharing yours. Two people in one login means notes in the wrong record, reminders from the wrong person, and no way to tell who did the appointment that went wrong.