If any email address is listed only for a person, then any emails from that email address will only appear in the timeline for that person if auto-shared.
Therefore, for an email to show up on an organization timeline, the email address must be added into their contact details.
Once the email address is added in, if it belongs to another contact you will receive a modal that asks you to choose the primary email address. Whichever contact is chosen will receive communications automatically if you have your email sharing turned on for that contact or for all contacts.
Because there can only be one primary owner, you will have to manually add the email to the other timeline.