In My boards, locate the Board you want to assign, and click the ...menu.

Select a user to share with and check the Assign check box...

The Board is shared as indicated by the message and the blue sharing icon on the Board.

On the recipient side, the (shared) assigned Board will appear on top of the My boards list.

If you click on the Assign icon on the Board...

..you will see a message that states that the Board is assigned to you.

In the Board details, you can also see that the Board is assigned.

Sharing - assigning Boards