We're talking about the shares that member banks buy from District banks; let's look at the law.
Banks buy shares at "$100 each" (from FRA Section 5). Now, when most people buy stock shares in a private company they may want to control or own the company and sometimes it's for capital gains or dividends. FRA Section 5 specifically forbids these goals by flat out saying that "Shares of the capital stock of Federal reserve banks owned by member banks shall not be transferred or hypothecated" and the fact that Congress even retains control of management.
The only reason banks buy shares (re- FRA Section 4) is so they can "avail themselves of the advantages of this Act." OK, so it's not much compared to what can be had with other kinds of shares, but there are a lot of banks happy enough with the deal.
The stock pays a 6% dividend per year by law.
... and the fact that Congress even retains control of management. The only reason banks buy shares is so they can "avail themselves of the advantages of this Act."
Each of the 12 regional bank districts is operated by a 9 member board of directors. 3 of the members are appointed by the board of governors of the FRB and 6 are elected by the member bank that own stock in the regional bank. Each District has at least one branch (most have multiple). Each branch also has a board of directors. The branch board of directors are appointed by both the FRB district bank (they appoint the majority) and the FRB board of governors (they appoint the remainder). These directors are involved in formulating monetary policy. While true that congress could over-rule anything the FED decides to do, it generally doesn't.