A partnership allows two or more people to run a business together, jointly make a profit and divide the proceeds in an agreed way. You should have a partnership deed describing the terms of your arrangements, although not having one does not prove that you are not in partnership; how you run your business and how outsiders perceive it, determines whether you are working as a legal partnership.
In dentistry, partnerships tend to be profit-sharing or expense-sharing. Each arrangement will have its own obligations and responsibilities; no two partnerships are the same. You should understand the implications of working in partnership and jointly owning and running a practice, the legal status of your working arrangement and the safeguards that you can introduce. You should seek professional legal advice from a solicitor experienced in dental practice and partnership law.
The alternative model of running the practice through a limited liability partnership (LLP) is covered in BDA advice Companies and limited liability partnerships.