Here is the list of the current dog beaches. These are cordoned off parts of the beach, never the whole beach itself like in France or England for example.
The closest are the following two beaches  at about 20 mns and 35 mns ‘ drive respectively  from the villa :
Badesi : Baia delle mimose. Go through the complex. It is the second beach after the barrier (or the man collecting 2 euros a day to park in the high season. You will be fined if you park nearby without having displayed the ticket in July / August.)
Isola Rossa : Park by the Spanish tower . Follow the path right of the tower to the rocks. There is a small gravely like beach if you carry on a bit to the right of the 1st rocky inlets. Park at : 41° 00′ 51. 21” N  8° 52′ 22. 12” E  or  41° 00′ 55.301” N  8° 52′ 29. 30” E
The following 2 are less than 1H00’s drive away :
Sassari : Fiume Santo (after Porto Torres, near the power station “Fiume Santo”)  40° 50′ 46. 67” N  8° 18′ 42. 83” E
Sorso : Discesa a mare number  4.       40° 49′ 10. 03” N  8° 30′ 42. 83” E
These final 2 are about 1H30’s drive from the villa :
Santa Theresa di Gallura ( 5 euros entry) : Spiaggia di Porto Quadro ( Turn for “La filetta / Terravecchia & follow the indications to Porto Quadro)
Palau : Localita “Punta Nera”     41° 10′ 53. 16”N  9° 23′ 32. 80” E
Go to the Tempio cascades too. It is a beautiful place and a great swimming opportunity for your dog too. See the blue folder for instructions . More tranquil before 1 pm generally.
More dog beaches get added every year so watch this space !
Beware though that the local countryside police “la forestale” do fine dog owners on non authorised beaches. You are probably safer on non authorised beaches early morning or late evening if they are virtually deserted. Out of season is mostly fine too (between October and early May , when the beaches are near deserted)
Dogs are otherwise allowed pretty much everywhere else ( lake, mountains paths, open countryside).
Quite a lot of Sardinians have a dog phobia. If yours looks big or threatening, you’ d better keep him on a lead around people. Actually, some are scared of labradors so go figure !!