What boat have you got?
There are some private islands and stretches of harbour front, but overall few places are "off limits". As above though, you do need to be conscious of depths and state of tide - Poole Harbour is very shallow in many places!
We often anchor, swim or beach the boats just off Brownsea Island at the western end (Pottery Pier) or the southern side just south east of the scout camp, both very sandy. A nice spot between Long Island and the Arne RSPB reserve, although not on LW springs. Years ago we used to anchor the old RIB just off Arne inland of the waterski zone. A nice anchorage around Goathorn Point. On a calm day you'll see boats pull in to the bays on Sandbanks just outside the harbour to the east. Or just to the west on the first stretch of beach between the chain ferry and training bank. Or, as above, go slightly further out, round the training bank and into Studland Bay itself, nice spots at South Beach and Middle Beach. Or further round Old Harry to Swanage Bay.