The position of the button depends on the use case and whether or not your toolbar is a global bar or specific to the current view of the app.
The left side of the toolbar is typically reserved for actions associated with the current view of the app. In the use case you provided the toolbar is custom to this view of the app (only appears during profile picture view) and when selected causes an action associated with the current view (downloads the image).
The right side of the toolbar is typically reserved for direct user action (configuration/profile) or some sort of call to action (signup, continue etc). In either of these cases the action does not directly interact with the current view and changes state either by navigating away from the current view or opening up settings.
The current mental model users are familiar with when interacting with global toolbars is that the button on the far left brings you to the "Home" or Initial load screen (feed) and the button on the far right allows the user to see things directly effecting them (config, profile, updates etc)
If the button drives an action associated with the current view (ie download/favourite) place it on the left side.
If the button navigates away from current view state and brings there focus to something else (configuration, next article). Place the button on the right side