Cool find!, I always like finding figs locally,the problem is ID.Because most folks don't care they won't mind someone "trimming" their tree for them and picking the fruit that makes a mess of their yard if you ask.I don't know what these figs are,this late in the season it's very hard to say much. The "white" fig has an interesting leaf shape,very shallow lobes and no thumbs,whilst the "black" fig has totally different leaves,are there the dominant leaf shapes on the trees?.
As the tree is close I would take cuttings and keep an eye on the mother tree next season,I've got about a dozen I've done just that with.The figs you picked are probably main crop that didn't have time to ripen,and with the cold weather they are getting a bit ropey and the colour won't be right because they aren't ripe,they look good though,maybe earlier in the season or the brebas will be a treat.Maybe with some ripe fruit pics and pictures of a selection of leaves next season folks can try and pin this one down for you,I hope so,I've got a few I intend to post about as well,best of luck to you