I would surmise that the running user needs per the doc
For Partner Portal contacts
Delete on Contacts and
For Customer Portal Contacts
Delete on Contacts
Edit Self Service users
Be aware of the considerations for merging portal contacts
You can merge a portal contact with another portal contact.
You can merge Customer Community enabled person accounts with Customer Community Plus person accounts.
Before you merge contacts that include partner users and customer users, decide which type of portal user to retain. The user that you don’t retain is disabled. However, after the contacts are merged, you can enable the contact for access to the portal from which it was removed during the merge process.
You can merge contacts that are associated with various accounts by transferring the contacts that you want to merge into one account and then merging the contacts that are associated with the common account.