Holkham Beach

The beach itself is not private land, there are no private beaches in the UK. Everything below high tide level (excluding a designated harbour) is covered by the same law as open sea, it is public property accessable to all.
Of course to reach a beach you may need to cross private land (or arrive by boat). In many cases there are public rights of way (footpaths) to get to the beach. This is the case here, Wikipedia says "The reserve can be accessed by footpaths from Wells and the local villages", and you can see those paths on the map.

As regards the Royal Family
a) The probably relied on nobody knowing they were going to be there, it is an isolated spot and people weren't as paranoid about security then.
b) Who cares (I am a republican).

Public access is through Holkham Nature Reserve.
There are narrow boardwalks that take you through the woods and over the tops of the dunes, and stairs down to the beach.
An excellent place for birding, or birdwatching, as some call it.
You have discovered one of the places I had hoped to go "off the beaten tourist path" on my next trip.
Nearby is the town of Wells-Next-The-Sea, and not far is Hunstanton, a resort town with seafood restaurants, a pretty beach, and hotels.

By the way, Holkham Beach is the location of the very last scene of the movie "Shakespeare In Love" with Gwyneth Paltrow, in which she is walking on a beach.
About a year ago, I researched filming locations used in this movie, and that's how Holkham Beach came to be put on my itinerary for next time.
There are a good many fascinating places to visit in the county of Norfolk. Norwich is on my list for next time.

Holkham Hall is a part of the estate. The grand family home is open several days a week for visitors.
It is possible that the Royals are the house guests of the Earl of Leicester (still his home) at Holkham Hall when they wish to visit, and are granted access to his beach.
To answer your question, "Or would they visit beach property that's within the privately owned Holkham Estate?"
My guess would be "Yes".

The Queen has her own country house and estate in Norfolk, Sandringham House (to the southwest of Holkham Hall).

Rebecca, fascinating information! This could be your tryout to be the newest guide book writer for Rick.

Adding to my ever-growing list!

Yes, I agree about this fascinating information -- thanks!

I too have been attracted to Norfolk as a beautiful destination that's not overrun by the tourism industry. And I love beaches; always keen to visit the shoreline in new (to me) regions of the world.

Pam and Epitd, thanks for your nice comments.
(I think that's one way of saying my posts are too long, haha!)
Epitd, you are welcome. Since you seem to be getting interested in exploring Norfolk, let me recommend a great travel guide to the area. Time Out Norfolk and Suffolk. (ISBN 978184670185-6)
It's my favorite guidebook I own. Lots of maps included.
This is the only guidebook you will need for this area, as it is very thorough.

epltd, I don't know if you do FB but I've been following a birding company in the Norfolk area for a year or so. I am a casual birder but like their pictures and like to get a sense of their weather, the scenery, etc. I've not been to the area and am more of a casual birder than someone on their tours but it's fun anyway!

https://www.facebook.com/The-Bird-ID-Company-355925021149845/

Very cool. Thanks Pam. Some interesting birds pictured there.

An excellent place for birding, or birdwatching, as some call it.

I've only ever known it as birdwatching, never heard it referred to as birding. Also twitching is used a fair bit in the UK.

Good old Wikipedia explains it all!

https://en.wikipedia.org/wiki/Birdwatching