Can I make private membership purchase links

Yes. You may want to hide memberships from your client page. You can do that in membership settings or when creating/editing the membership.

Toggle "Show on my client page" off to hide the membership from your page.

Clients can still purchase the membership if you send them the private link. Under Membership Settings scroll down to "Private Membership Link" and you'll find a link you can give them.

Don't forget you can also add members to memberships manually.