You're taking me back to the past with this one.
Back when I was still living at home with the folks and indulging in secret, I would usually dispose of my diapers in one of two places, the dumpster behind a small strip mall near my place or the garbage cans at a local park/park restroom. I always made sure to double bag my diapers before dropping them off and if I could I would dispose of them late in the evening or in the wee small hours of the morning so no one would catch me.
My house was just a block away from a strip mall and there was a number of dumpsters behind it, that was my usual drop off place. However, another block away from my house was a local park, the park had numerous trash cans and a washroom pavilion, I would usually dispose of my diapers in any of those. Otfen times if one of those two locations was occupied, I would just run on down to the one that wasn't. After I had bagged the diapers, I placed the bag in my backpack, it would look odd if I was walking to the park or a dumpster with a trash bag in tow.
The best method in your situation is going to be the method that is available to you. I don't know where you live, but if there are no dumpsters or public trash bins open to you, then you might have to try and sneak a bagged up diaper into your regular trash bin just before pick up, of course, you'd have to time things properly for that to work out.