Computer should not be sharing an IP Address. Each machine will need its own IP address so that the computers now how to talk to each other.
If one computer is on, and your router reboots then another computer is turned on then both machines could be issued the sameIP address. Because of this it is recommended to not reboot the router (or other machine which issues the IP Addresses).
IP address conflict will cause problems to networking applications and resource sharing, but privacy won’t be affected.
Sometimes Windows computers get “sticky” DHCP addresses that cannot easily be released. As a previous poster suggests, turn off the router or access point for a while. On each computer, if XP, click Start, Run, and type CMD. At the command prompt, type <i>ipconfig /release</i>. This will release the current IP address and network connectivity will be lost at this point. Turn back on the router and wait until it is completely online. Then on each computer still at the command line, type <i>ipconfig /renew</i>. This will assign a new address to each computer. If both continue to get the same addresses, then contact the vendor of the router device to find out if there is a problem with the DHCP services or if a firmware upgrade is needed.