I have never been lucky enough to go during the free dining promotion. We still get the dining plan, because it is worth the cost if you use it right. The deluxe plan works the best for us, but I say you get whichever one works the best for her. If you are not planning on eating at many table service meals, get the quick service plan. Technically, you are not supposed to share (and we get the deluxe, so we never need to share). However, you are welcome to eat anything that your mom cannot finish on this plan. That is better than throwing it away. The mug can only be refilled at the quick service location in her hotel. There are a variety of drinks you can put in the mug. It usually excludes milk, juice, alcohol (of course) and some others. My son only drinks milk and bottled water, so his mug is basically a waste (but it comes with the deluxe plan, so we get it).
If you get the regular dining plan, you are still welcome to do the same as above with your quick service meals. Table service is a little bit more tricky. If you are eating at a family service or buffet, she obviously cannot share with you. Everyone would need to pay for themselves. If it is a sit down table service meal, you are more than welcome to order an appetizer or something small and eat whatever your Mom cannot. Disney does not charge plate splitting fees that I am aware of. The wait staff will not likely discourage this. I do suggest you order something (even if it is very small) since you are taking a space in the restaurant. That gives the wait staff a little extra in the form of tip. It would not be fair to take that space if you do not order anything in my opinion.
You can pay separately from your Mom or just have them deduct her dining plan meal from your bill. For tipping purposes, you will see what the entire meal would have cost if she did not have the plan. That way, you can tip the right amount. I always order deluxe, so I only know that costs $78.99 per adult and $21.99 per kid 3-9. I would imagine the regular plan and quick service plan would be a bit less. I read somewhere that the difference between the QS plan and
DDP is only about $10 a day per adult. If you will eat at least one TS per day, it is worth the extra $10 to get that plan.