By "Chechen fighters" I mean those Chechen residents who took part in the fights in that area.
AFAIK, Chechen fighters are as omnipresent as American mercenaries in the Ukraine.)) Everyone knows about their existence but is still unable to capture these elusive creatures and demonstrate their true faces to the whole world thus far. So, it’s safer to consider all them byproducts of information wars for now.

I doubt the civilian people there support any of the sides of the conflict. What it really is is some masked a**holes who want to play a war vs. regular Ukrainian army. It's not a surprise that the latter should demonstrate a bigger ability to smash the enemy than the former one, which is eventually going to happen.
You draw these conclusions from popular American press, don't you? I read NYT from time to time and find its coverage entertaining. But the editor-in-chief was fired the other day, as Putin would say, God knows why?

В американских СМИ активно муссируется и другое предположение: увольнение Абрамсон может быть связано с появлением в газете 3 мая репортажа из Славянска. В нем журналисты, которые провели несколько дней с пророссийскими сепаратистами, рассказали, что среди них нет граждан России, а сами они в большей степени представляют местных жителей, часть из которых — бывшие военные украинской и советской армии.
Главный редактор The New York Times Джилл Абрамсон уволена со своего поста - Газета.Ru | Политика