Is your peer-reviewing service ever important for career advancement? Recently, there have been a couple of active questions on Publons (e.g., Publons - is it a serious thing? and Is it worth creating a profile with Publons?).  One of the supposed selling points is that it allows people evaluating you to verify the peer review work you've done.  At least one user says your peer-reviewing is one criterion used for promotion.

I am in pure math at a public research university in the US, somewhat senior, and I don't recall ever having seen any real discussion of a job/tenure/promotion candidate's peer-reviewing activities in the evaluation process.  It is a given that you do some refereeing, and it counts as part of your service to the community, but unless you state that you do an exceptional amount of refereeing or consistently referee for the very top journals, I probably won't even notice your reviewing activities.  If I do notice, I will quickly forget.  (I don't list the journals I review for or how many reviews I do on my CV, and I believe this is reasonably common in math.  So I wouldn't even know if someone I'm evaluating isn't doing any reviewing.)

However, the nature and expectations for peer review for pure math seems to be quite a bit different from other fields.

Question: Are there departments that, say for hiring, tenure or promition, use candidate's peer-reviewing activities in a signficant way in the evaluation process?  If so, what kind of departments use this and how?

(While "significant" may be somewhat hard to pin down, two examples are: peer reviewing regularly gets discussed when evaluating candidates, or if there is some formula for numerically scoring faculty which incorporates peer-reviewing activities.)
I do not believe simple refereeing activities have an impact on hiring or tenure, but it may have an positive effect in an annual report, to get “progress through the ranks” salary increases or bonuses, and other recognition at this level.  

It could have an impact in exceptional circumstances, such as receiving a referee-of-the-year- type recognition, but it would be truly rare for a junior person to get such recognition and thus have weight in a hiring decision.  For a more senior person repeatedly getting such recogniztion might be useful in a promotion file.

Doing a good job at peer-review is usually a stepping stone to getting an associate editorship or a senior referee position in journals, and these are often considered in tenure/promotion decision as evidence of oustanding reputation beyond the institution.

Note that outside public recognition of work done as referee or on an editorial board, rather than the anonymous work usually done as a regular referee, is usually what is prized.

I do not have a Publon account: I am tenured and consider it part of my job to contribute to my field by refereeing the work of others, just like I expect others to review my work.  I do get the occasional accolade, which I will gladly take, but  I don’t do referee work (papers or grants) for recognition any more than I give to charity for recognition.