Link to send message is not displayed
DrupalGap's implementation of Private Message has not evolved enough to take into consideration any Private Message configuration set on a DrupalGap Manage Display page for User Entities.

That being said, I would recommend creating your own page to custom render a User Entity. With these docs, essentially replace the usage of node with user and that'll give you a custom page in which you can render your users.

Then in your page_callback function to render the users, you can add a widget to the content render array, which could display a button that when clicked will navigate the user to a custom DrupalGap form+page where you can build the input elements (recipient, subject, message, etc).

Then in your form's submit handler, you'll actually send the message to the user.