It depends on the beach. Common opinion is that all beaches between mean high water and mean low water (the foreshore) belong to the Crown but that isn’t the case. Some beaches are privately owned so the owner’s rules may apply. I fly on a beach that is owned by the local sailing club but I yet have to be approached by a red-faced and angry committee member. I get a lot of interested passers-by wanting to know about the aircraft though.
The Crown is fairly sensible when it comes to flying :
"Drone flying / Unmanned Aerial Devices (UAD)
We grant permission for UAD flights over Crown Estate foreshore (defined as the land between mean high water and mean low water).
This permission is subject to all operators complying with the applicable laws, statutes, regulations and codes, as well as the requirements of the Civil Aviation Authority, and obtaining any other necessary consents for the operation of UADs.
On occasions, third parties such as Government agencies or local authorities may restrict the flying of UAD’s on Crown Estate land. We therefore recommend checking with the appropriate authority to ensure that no such restrictions are in place."
To find out which areas of foreshore we own, please use our Foreshore and Estuary Ownership Map
Once you have taken off from a beach you aren’t confined to staying within the limits of the foreshore.
My understanding - I am NOT a lawyer - is that airspace above a property is administered by the CAA and, as long as you are following the Drone Code and not invading anyone’s privacy you are flying legally.