Is it obligatory to dedicate supplication [du'a'] to the person on whose behalf the 'Umrah or Hajj was performed, or is it permissible to supplicate for oneself and others?

You are not required to dedicate every supplication to the person for whom you are performing Hajj or Umrah. Rather, it is better to supplicate for both yourself and them. If you supplicate for yourself only, there is no harm in that. The reward for the ritual act returns to the one who commissioned it, and it is commendable for the proxy to include the one who commissioned it in their supplications, such as saying: "My Lord, forgive me and my brother." However, if they supplicate for themselves only, there is no harm.