like your memorable username 
I think Bitshare promises better anonymity than Bitcoin, IIRC.
It does .. but the forum still logs your IP which are viewable by administrators 
Currently BitShares anonymity is similar to Bitcoin, just easier to use. The wallet as currently implemented goes the "easy route".
1) You get a new address with every transfer
2) No change addresses are used... so you can tie transactions together this way.
3) If you are voting for a unique slate then that can also tie your transactions together.
4) If you receive funds from multiple people and then transfer them all at the same time, they will be tied together.
End result: privacy of the *CURRENT* wallet is similar to how bitcoin operates.
The blockchain supports greater privacy, but the advanced privacy features are not exposed:
1) Use change addresses
2) Never combine inputs, instead use multi-transaction payments
3) Reuse existing slates of other users (or use recommended slates)
4) Pay USD transfer fees with BitUSD