Generally speaking you now have two choices:
If you decide to keep the old website, make SURE that all associated accounts are transferred into your name (google account, hosting, domain registration, etc). And make sure that the website is up to Google standards and is modern (mobile responsive, etc). If you are making major changes to the site, or completely rebranding the practice, it might make sense to build a new website.
Often the best choice is to build a new website. If you do elect to build a new website, first make sure that in the purchase agreement you are given ownership of the current website and the right to use the owner dentist's likeness, name, photo, etc. for branding. You do NOT want the owner dentist to disappear completely from the marketing picture -- especially if he or she marketed their name.
If you are creating a new website, use the existing website to help capture patients from the existing practice. If the existing website is well done, and ranks well, consider 301 redirects to help preserve your search engine rankings. But if the existing website is going to be scrapped, plan to scrap it but not just yet!
Instead of scrapping the existing site, remove the pages other than the home page. Change the home page to reflect that the practice has been purchased, and provide a link to the new site, and be sure to include the outgoing dentist's "patient transition letter". Remember, there are patients out there who haven't been in for a recall in years, or who broke a denture and will now try to find "Dr. Smith". If they Google "Dr. Smith" and you come up, they won't understand what happened and there will be a disconnect. Instead, if you have the existing practice's home page co-branded, with an explanation (the letter) that Dr. Smith is proud to have sold to you, Dr. Jones, then these potential existing lost patients will more easily reappear.
In this manner, you have Dr. Smith's site for Dr. Smith's patients and patients of the practice who did not know about the sale/transition, and your new website (holding website at this point) to start enticing new patients!
At Short Hills Design, we recommend that our clients put up a temporary site, or holding site while their new website is being constructed. You do NOT want current patients, potential new patients, or previous patients of the practice to Google you and find nothing. You can review this article for a list of what should be on your temporary website.