It doesn't. The only way an ice sheet gains mass under any normal circumstance is new snow falls on top. As more and more snow accumulates over time, the ice sheet grows thicker.

What offsets that is that ice flows under pressure, so the entire sheet has a tendency to head for the ocean. Things are a bit complicated in Antarctica since some of the landscape where the sheet is grounded is below sea level. There are concerns that the seawater is getting warm enough to cause melting there, increasing the flow rate.

If, in fact, ice sheet mass is increasing, that's great in terms of sea level rise. There's no other way to spin it.

You have a layer of ice covered with a layer of snow. Each year, that snow melts a little and then re-freezes on top of the ice. Then, more snow falls on top of that snow. This happens year over year for a long time. Each layer of snow puts more and more pressure on the previous layers of snow and ice, compacting them. Eventually, these layers of snow become so compacted that they form new layers of ice and fuse into the existing layers of ice. Does that make sense?

I assume it means snow melts, turns into water, then freezes again turning into ice thereby increasing the mass of the ice sheet.

the density increases not the mass. the concentration of mass is increased density. it is further evidence of melting.

It reads to me like there's a distinction between ice and snow. Like the snow isn't considered part of the ice or isn't able to be measured as part of the ice until some compaction has happened, and the "snow" layer might be hundreds of years worth of snow. That's the only way I can resolve the definitions with the statements, and I'm hoping someone can more authoritatively describe the paradox because I'm feeling unsatisfied.

there is also a distincion between the coastal ice sheets and the continental interior ice and snow cap. the apparent paradox is caused by misrepresentative interpretation of a report, by the author of the article.

the coastal ice shelf and continental interior are two distinct physical systems.

isostatic rebound is a confound to the measurement of snow/ice accumulation when change of altitude is corelated to snow deposition.

http://rses.anu.edu.au/geodynamics/tregoning/7.pdf [PDF]

https://www.worldatlas.com/articles/what-is-post-glacial-reb...

BTW Re: " and I'm hoping someone can more authoritatively describe the paradox because I'm feeling unsatisfied."

you wont get anymore of an authoritative take on it than discussing it with a scientist, as you are currently doing right now.

That really isn't how mass works.