Please don't throw out your dog's poop here. It's not very nice for the rest of us. It's not a public trash can. Please don't throw out your garbage from home here.
この場合、次のような言い方ができますよ。
ーPlease don't throw out your dog's poop here. It's not very nice for the rest of us. It's not a public trash can. Please don't throw out your garbage from home here.
「犬の糞をここに捨てないでください。私たちは困っています。公共のゴミ箱ではないので、家庭ゴミをここに捨てないでください」
to throw out ... で「…を捨てる」
a public trash can で「公共のゴミ箱」
garbage from home で「家庭ゴミ」
ご参考まで！
カナダ 
