I use brine from both sweet and dill pickles and other pickled items in cooking. There's lots of flavor in them as well as acid and salt and adding a little to a soup, stew or sauce can popup flavor. Best of all it's already paid for.
The jar of pickled cherry peppers in my fridge is the most coveted for this purpose and it takes some disciple to leave enough brine to cover the peppers.
I use a little, as well as a little finely chopped half sour pickle when I make corned beef hash for Craig. We picked apart the hash from his favorite breakfast place and figured out that's what they did.
I use some pickles brine when making some rye bread - especially dill pickles, which make it really good. The original recipe I saw using it had 1/2 c per 2 c of liquid, but I use 1 c, or 1/2 of the total liquid.
Sometimes, usually in the summertime, I'll keep the juice from commercially-produced dill pickles and immerse thinly sliced fresh cucumbers in the juice to make my own, um, dill pickles. Pretty good if left a while in the refrigerator.
We usually use the pickle juice, either by adding more veggies once the pickles are gone (carrots, peppers, green beans, cucumbers, etc.) or by using it has a meat brine/marinade. Bread and Butter pickle brine is great for a pork marinade, gives it a sweet and sour flare. Dill pickle brine is good with chicken but also a little bit is good with fish too. Sweet pickle we don't use as a marinade, just add more veggies to it to pickle them. Pickled pepper brine is great for just about any meat. Also can be used in place of straight vinegar for hot sauces.
Next question about using leftover brine to pickle other veggies - how long does it usually take for the veggies to taste pickled? I'm thinking of trying with the brine from fermented pickles. If it doesn't seem to work, I'll add some vinegar. But, I want to have an idea of how long before I decide it didn't work.
Usually about a week seems to be good but it depends on the vegetables. Things like peppers, cucumbers, green beans that are spongier, less dense are sometimes good in three days depending on how thickly they are cut. Carrot slices, radish, celery, parsnip are a solid week. Onions are two days to two weeks depending on how raw/crunch you still want them to taste. Dense veggies like broccoli and cauliflower typically take about a week and a half to two weeks before they soak in the flavor.