How concerned should one be if he/she does not get referee invitations? A friend of mine is an experienced researcher in early career.  We are surprised to learn that she never get invited to referee for any journal.  I once rejected a referee request and referred the editor to her because the manuscript matches better with her expertise, but still, she did not get invited.  I recently learned that this happens more than once: some one else also recommended her for referee but the editor did not take the advice.  It becomes a little bit concerning: is she somehow blacklisted or what?  What is really happening here?
It's quite possible the editor never read the recommendations.  Some journals invite more reviews than they need.  If some reviewers decline, but enough reviewers return reviews for the editor to make a decision, then the editor will not consider inviting more reviewers.  If the decision is to reject, it may be made after only one review is returned.
Never getting invited is a bit odd, but could occur by chance.  I only get 33% of my fair share of invitations, even though my reviews are faster and more detailed than the other reviewers.  Most journals in my field have never asked me, other very similar journals ask repeatedly.  I have received no invitations for many months, and then gotten two from different publishers on the same day.  Don't overanalyze it; don't be concerned.
Most publishers have a form you can fill out indicating your research fields.  If you complete this it might increase your chances of getting papers to review.  In my experience, it is not necessary to publish with a journal or publisher in order to get asked to review.