Same here. I work for a small family owned GC and we do mostly rough and finish carpentry, with some oddball stuff mixed in here and there. They are more than willing to supply power tools; battery and hand tools are on us.
However, I prefer to use my own tools, and he is willing to repair or replace, and pays for all consumables....blades, nails etc. I'm really picky about my tools and am willing to pay substantially more for quality to make my life easier. If he was to buy a mitre saw, I'd probably get a harbor freight special. So, 95% of the tools in my van are owned by me, including compressors, saws, nail guns etc. The only tools company owned are the rarely used ones.
The other end of that is no one else uses my tools. They are mine, and I take care of them, make sure they are put away neatly, clean and in working order. If another guy needs a mitre saw, they are free to use the company cheapo or their own. A couple guys I work with abuse their tools and/or don't know how to correctly and safely use them. Drives me absolutely nuts so they don't go near my sh!t!
But, I've been in construction for 20 years and have amassed a sh!tload of tools that would otherwise just sit in my garage. I'm more than happy to use them. Some of the younger guys starting out, who haven't had the chance or the funds to purchase an entire tool store, might need some help from the employer. In that case, I would say the employer should provide anything with a cord. Battery and hand tools are employee responsibility.
The company I work for will supply all major power tools. But I still use most of my own. I just like my stuff. He'd take care of any repairs if we asked, and we can pick up anything we need as far as blades and just charge it at a supplier.