My ex boss has a house at Rhoscolyn and we spend at least once a year there. A lovely dog friendly house with a big garden.
When we are there we just literally step out of the garden and have a very short stroll down a lane to a quiet (during the winter or weekday) beach, the water is shallow so not dangerous for children or dogs and the water is so clear, the sand lovely and clear with some rock pools too. There are basic loos there but its the sea, sand, wildlife and good ol' fresh air that makes it a wonderful place. Great fishing off the point too

All the time we have been there, there are no restrictions of dogs on the beach.
In the summer, the pathway east is lined for about a mile with very very fragrant wild rose bushes. I can smell them now

Wonderful
You can walk out along the west or east coastline for miles and the view across the Lleyn Peninsular are breathtaking.
It is a little bit of a pain to drive to as the lane is very narrow but its worth it
MapJust take a look at the hills in the distance (taken from www.anglesey.info)