Access to other Sites
Your default route is probably the reason you can’t access other sites in site B. Check that your gateway switch or router has IP routes to the networks off of Site A. A very unrestricted (kind of frowned on..) route would be “IP ROUTE 10.0.0.0 255.0.0.0 (site b pix IP)”
Outside Access to Site B
It sounds like your access lists are restricting access to the VPN by address. Meaning only clients with an IP address in 10.10.0.0/16 will be able to access site B. If your client VPN assigns a DHCP address to the clients, make sure that address range is included in the access list.
Sorry, we need more help on this!