'Do people, just because there are no dedicated bins by the entrances, think it absolves them of the responsibility of acting as a good dog owner and being a person who thinks of others?'
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YES.
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When my bro and I, along with little Gaylord, stopped on a site in Kent the other month, the site owner said, 'feel free to walk your dog through the miles of open countryside, as we own some of it and rent some of it' (or something like that). She did request that you picked up after your dog even in the fields.
Guess who was the only person to walk back onto the campsite with a little black plastic bag? Me!
I live next to the Grantham canal, and it's the same along there, dog muck all over the place despite the fact there is a plentiful supply of dog bins. Spoils it for everyone else and gives all dog owners a bad name.
At the site in Penrith, I was walking to the onsite shop with Diesel when he had an accident on the edge of someones pitch (he was on a short lead), I won't go into the details but it was similar to your dogs predicament. I cleared most of it up but then, well you know, so I rang my brother and he came across with a large bottle of water and I managed to wash the rest of it away.
Lisa