I don’t think you can see on the user’s timeline whether a tweet is promoted or not. If it is a promoted-only tweet then it will not show on the user’s timeline, if it’s just a promoted tweet it will but I don’t think twitter marks it in any way so as to know that that tweet was promoted by the user.
You can request information about a tweet using https://dev.twitter.com/rest/reference/get/statuses/show/%3Aid
I guess you could infer something from the
scopes properties of the returned object. For example if there is a
scopes property like
public 'scopes' =>
public 'followers' => boolean false
then you would know that that tweet is a promoted-only tweet, because it doesn’t appear to the user’s followers. But I think that’s about all that you can find out.