What to do with cottonwood stumps?

We have two cottonwoods growing at the edge of our property and for years we have known at least one needed to come down.  One grew bent and is now over the house so time for that threat to be removed. The other tree is upright but both trees are growing so close(almost merging at ground level) that I think this one will need to go also just so they can take the leaning tree down.  

We know they have to be removed for the safety of the house and of anyone that drives past them in the alley. Big question is how to treat the stumps.  Have them ground down, leave them a bit above ground and let them regrow? Leave a bit taller, girdle base and get some mushrooms for them?   The trees are in a line of Bridal Wreath Spirea  that have been there at least 30 to 40 years so I am wondering if this could be a nice place for a little bird bath or insect watering station... OR could I actually do both a watering station and mushrooms?

YES I will be asking them to leave the chips here for me to use as mulch  :) I mean I am sorta paying for them right?

Living in rural southern oregon, I have seen cottonwood tree blow downs, that sprout out oyster mushrooms, naturally. You can just bore holes, make cuts etc...and mushroom spoor may appear naturally. You can buy spoor inoculate, and nowdays you can actually get chainsaw bar oil, that has mushroom spoor in it. I also see oyster mushroom on alder species. Oyster mushroom is one of the most common wild edible mushrooms.

I think I will go with one of the oyster mushroom species for the stumps and possibly put in a small diy bird bath on the upright stump IF they can leave it long enough for me to do so.  The two trees are almost conjoined at the base so that may not be possible.  
I was surprised one company did come out and look over the trees and pleased with the price they quoted.  Not as happy with the fact they leave it in logs that they haul off.  However he did say for another $100 I could get a pickup load of chips for the garden...  Will have to think on that,  maybe check out the not so local nursery in Colorado for wood chips.  On the other hand I am not sure what is and isn't open down there.   I do have one more company coming to look over the trees and give me a quote so it is all on hold until then.

I can't speak precisely to your species of poplar. The ones I've had on my property offer incredibly tough, long-lasting and resilient stumps, hold-fasts, call them what you will. They are magnificent chopping blocks, possibly foundations for semi-permanent sheds, and heaven knows what else. Simply stated, they are a resource -- if you can find some way to utilize them.