To do this you need to communicate very clearly why are you doing it, and if it will have any influence on performance reviews. But I do think that monitoring estimates in this way will affect the quality of estimates.
A manager in my previous company noted everyone's estimations during sprint planning in an Excel sheet. No-one knew why, so I found myself thinking more about that Excel sheet than about making an well-founded estimation.
(edit: of course then I'd have to fire half the team again... and again... until only one or two incredibly depressed pessimists remained... tough call)
I never understood why managers get so upset over consistent time under estimates, just track it and add the relevant factor without telling the team and it should on average be on point.