I've never had any training, but I only carefully chainsaw small branches off trees, or trees which have fallen over, or logs which have already been cut by tree loppers. I don't know how to bring down a full tree.
The other factor is the chain, where plainly accidents can happen if the chain breaks while being used.
I always pay the local chainsaw and mower repair place $9 to sharpen each chain. One could buy a chain sharpening kit from Bunnings and do it one's self. But the guys who sharpen it in the store tell me when the chains have had it due to wear and need to be chucked out. On one occasion, because of weird wood I had been cutting (old, very hard on outside, soft inside) one new chain had had it by the time it was due for its first reshapening, whereas other chains have had up to four or so resharpenings before throwing them out.
Regards,
Renato