Yes they will. Because of the dogs I let mine compost first. Sometimes they go into a flower bed the dogs don’t access but not onto the lawn straight away.
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omg…This is starting to take on the tone of the dreaded toilet paper dissertation.
LOLOLOLOL!!! 
Yes, they are compostable, and we do encourage guests to compost. Our last guest asked me exactly where the compost bin was. But a lot of folks don’t know how or don’t want to bother on vacation (I’m lucky if they’ll recycle, like others here I do a lot of picking things out of the garbage). I personally find the mesh filters to be a pain in the you know what cleaning wise, so I won’t put one in there for guests. I’ll just continue on my search for the perfect container. 
I’m with those who find the wire filters too messy to empty and wash.
I store the guests’ paper coffee filters in a short, wide glass canister that has a metal lid. It matches the taller glass jar filled with their ground coffee.
I bought them both at a dollar store.
There’s a wee note on the coffee-making centre asking guests to put the used, filled filters in a bin, separate from the garbage one. Most comply.
As the coffee filters are unbleached paper i then dump the whole thing under the rose bushes and once in a while till them into the soil.
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I keep mine in a drawer along with the tea towels just under the coffee maker and it has a label on it. No one has ever asked me where the coffee filters are.
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No I don’t like to clean. Also, I’m usually the one making the coffee and it’s a breeze to dump the grounds and rinse out the basket.
Well then, you’re welcome to come and make the coffee and use the mesh basket. I’ll leave the door open for ya’.
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I’d suggest, if you want the filters in plain view, you could get some pretty paper (Michael’s has a huge selection or print it out… not that much needed) and cover the box with it. It would be cheap , easy and add a decor type element. Then if you wanted it to last longer, cover it again with clear sticky paper printed before peeling with “filters”. There are also numerous camping type small metal containers that you can find on etsy or ebay if you want something very durable and hard to open.
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