The problem is, "breathing problems" have many possible causes.
I've known people who had skating-related allergy problems, attributed to mold. (Almost any surface on which water condenses can grow mold. Add a few hot lights, and conditions are perfect for growing mold.)
They say it varies a lot by rink, and by season. But it should be worst in the summer - some rinks don't have good dehumidifiers.
An outdoor rink might actually have fewer problems, because they get better ventilation.
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As a separate issue, I got a little sick in a rink which I think uses a gasoline powered Zamboni, when I was there most of a day. Depending on the condition of the Zamboni (and edger, if used) and how much they ventilate the rink, you can get fairly high levels of Carbon Monoxide (CO). Some fire departments and county health departments are willing to test public buildings for safe CO levels.
A natural gas powered Zamboni could create significant CO too, but only if it was poorly cared for. I guess (no data to support this) an electric Zamboni could create significant Ozone (O3), if it was in incredibly poor condition - but that is mostly only a problem for people with asthma.
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Really, really tight clothing could create problems too.