How does the brokerage sending me .60 cent cheques know I've gotten rid of it?
If you have the shares at home and they are registered in your name, then there is no brokerage sending you money. The money is paid by the company to the registered owner (you), as recorded by their transfer agent.
If you take the shares to your bank's brokerage, then you will transfer the ownership
from your name to the broker's name. Then the dividend will go to the broker as the registered owner, and they will credit the money to you as their client. When you ask the broker to sell the shares, they will transfer the registration from them to the new owner. The company's transfer agent takes care of the transfers and records the ownership.
A lot of people are unaware of the fact that when they buy shares via a discount broker that they do not actually own the shares. Their broker does. The broker just records the shares as owed to your account. They actually hold your shares in their name instead of in yours and just give you credit for them in your account. It is easier this way because then they don't have to get you to transfer shares every time you sell. Since they hold thousands of them and buy and sell thousands every day, no shares actually trade hands. There is just an electronic change in the numbers. It is just the same as if you write me a cheque for $100 and I deposit it to my account. They simply reduce your account by $100 and increase my account by $100.