At tax time, I am overrun with 1099s. Sometimes the shares can be transferred directly to your new account, or you can request a stock certificate that you deposit in the brokerage account upon receipt.
Brokerage accounts are beneficial in many ways.
Costs can be minimal, depending on the services associated with your account. You can accumulate dividends, conveniently reinvest in your securities, or buy new securities as a means to diversify. At a brokerage, you can sell shares quickly, or set your own price and bide your time. When stocks split or if there is a spinoff, the brokerage account holds these new shares as well. It is common for brokerages to send monthly statements showing the value of your holdings and any activity in the account.