Generally no, but I think having a few pieces of equipment of your own indicates that you are serious about the job and interested in learning more. I have used my own saw because I got tired of being accused by a boss some years ago of running the chain into the ground....ergo if anyone's chain got dull for that reason it was my own. For the amount I used it no big deal but it was mine, not his And these days I always carry it to buck up any good firewood I want for myself. I also have my own rope and carabiners, and on a couple of occasions a more appreciative boss did request and make use of them as his was not quite long enough. I also carry my own tools, scrench, files etc.
Ditto for PPE-I always have had my own helmet combo, boots, gloves, glasses and saw pants, literally the only piece of gear I have borrowed on occasion was a saddle. I do not like wearing gear that is shared with others-personal thing, and given I have three saws of my own again no biggie.
It has been said elsewhere on this site many times, strongly consider buying Jeppson's Tree Climbers' Companion, it spells out in really simple terms a lot of the activities you will be supporting as a groundman, and if I can understand it , anyone can.
Good luck.