Yes, the recycling that you pick up comes back to your city for processing.
The process for recycling works like this:
Trucks leave in the morning and start collecting. When a truck is full, it returns to base, drops off its recycling, then goes out again. Trucks all return in the evening. (That there is some conflicting information about this, usually with cities that have not enough recycling capacity, but this is my personal observation)
When there at least 1 unit of recyclable material in the recycling center, it will process it into metal/plastic/alloy according to the reclamation modules you have built.
The Recycling Center's delivery trucks will move the processed resources to your trade depot/port.
Assuming you're in this for pure profit, you have your trade depot/port set to Export the resources. Your trade depot/port will export the resources via truck (or rail/sea/air if you have a properly equipped trade port).
If you're sending trucks to your neighbors, it takes time for the trucks to travel there, pick up, and return. Their return will be staggered enough to give your recycling center plenty of time to process the recyclables into resources.
If your recyclables storage is full, your trucks won't leave to pick up more recyclables. I'm not sure if you can lose recyclables due to the center being full when your trucks return, but this is a non-issue. The only time it could happen is if you don't have enough recycling capacity to match your city size. In this case, it doesn't matter if you lose recyclables, because you're generating far more than you can process.
With an educated city, especially a low/med wealth high population, you'll find that one recycling center is often not enough to cover your own city, and you'll end up building more than one. You won't be able to spare any trucks to cover your neighbors at this point.
- If you're recycling for pure profit, just fill your center with Alloy reclamation modules and set your trade depot/port to export. The other resources are not worth as much.
- If you're generating resources for a great work, set your trade depot/port to local use. You can also use import, but it can get very expensive.
- If you're generating resources for local use, set your trade depot/port to local use. You'll have to keep an eye on the depot/port storage; If you have 3 recycling centers producing alloy and plastic and only one processor factory, you're likely going to have a surplus in the depot/port, which will in turn disable your recycling trucks if the center's storage backs up.