it depends on the pension....some allow them to go to a named beneficiary if the pensioner dies but often they can't be inherited beyond that one named person. I find your question a bit confusing so will try and answer
Person A pays into a private pension. Person B is the named beneficiary of the pension if person A dies. Person A dies so person B receives the pension (possibly at a reduced rate) When person B dies, the pension usually dies with them.
BUT same person B also has a private pension....they had named person A as the beneficiary but person A died before person B died. There might have been the facility for person B to name another beneficiary for their pension, but not person A's, but I suspect that if this wasn't done then the pension dies with person B.
BUT it depends....different private pensions have different rules so whoever wants to know needs to check with the pension provider(s)