The user that you want to transfer the connections to needs to have edit access to both source and destination sheets that the particular connections are linked to.
You can transfer the ownership of a connection in three simple steps. This person will then be the new owner of the connection and can fully edit it.
First, select the connection.
Second, click on "Transfer ownership" in the top right menu.
Third, enter the email of the user who should be the new owner of the connection.