No, you don't, but if you do have a qualification or certificates, it proves that you can deliver properly put together letters and referrals, and that you understand all that's necessary about confidentiality etc. You need to be accurate and understand paper and electronic filing systems, and medical records.
If I were going to recommend a qualification to you, you'd need either the AMSPAR (Association of Medical Secretaries and Practice Administrators) certificate, which may run at your local FE college and costs about £2k over two years (one evening per week), or look at the on-line courses run by the BSMSA (british Society of Medical Secretaries and Administrators) at www.bsmsa.org.uk (about £800 for full certificate course). If you don't have the qualification then I would suggest that you need certificates demonstrating your typing speed, and some consultants still use audio tapes for dictation - shorthand isn't needed but something like Pitmanscript can help you write quickly! Some Med Secs in the area where I work also have the NVQ level 2 or 3 in Business Admin, to demonstrate they are working to national standards.
You might need also to study topics like Medical Terminology (on the BSMSA site) or Understanding Investigations, or Prescription Medicines explained - all to help your understanding.
I hope this helps!