Yes and yes.Both are right on the beach. No limit to stay/sit/lay on the beach whenever you want.

From both hotels, you just walk across the property to get to the beach, although there is the sidewalk path that must serve as the official boundary. I've walked along the beach between the Laguna and the Grand Hyatt. Along the way, there were a couple local very informal food places on the beach and a bunch of stands were people sold cheap souvenirs etc. Some of the hotel restaurants have outdoor seating overlooking the beach.

Although the beach itself is public property, there's no easy public access as the beach resorts in this area of Nusa Dua are essentially in a compound with no free parking or convenient public transportation. The Laguna has a guardpost at the road entering the property and the Westin has a similar arrangement. [There's a temple located on the boundary between these hotels, so I don't think local public access is totally forbidden.] IIRC there are also some hotel cabana type thingies on the beach, available for rent by hotel guests and clearly provided/staffed by the hotel. You won't generally see crowds of people on the beach who aren't hotel guests, but I think there might be some local festivals that involve everyone going to the beach on a few special days.

HUH? The Westin and the Laguna are two separate Starwood properties. They're next door to each other (the easiest way from one to the other is to walk along the beach) but they don't even seem to share any facilities. AFAIK you can't even do a room charge for F&B or gift shop items at the neighboring hotel.