NOTE: Entities may only be shared (not give up) with Limited Roles. You can still give to any role.
If you want to share entities from one organization to another organization on the same server, you first need to allow the sharing between the two organizations.
To do this, you need to open the Edit->Sharing Permissions screen for both organizations.

First select the Giving/Source Role and go to Edit->Sharing Permission.
Copy the full name of the Role which includes the Organization name between the parenthesis. (1).
Then go to the Receiving/Destination Role and go to Edit->Sharing Permissions.
Paste the copied name into the input field (2) and click Add.
You should see the Giving Role now listed below (3).
You can now log in as the Giving Role again. Select the entity you want to share, and below the entity on the overview screen, you should now see a share link. You can get more information here
To remove sharing later, you can tick the checkbox and click Remove (4). This will not stop sharing the entities that you have already shared, but it will prevent any further sharing.