In the USA, somehow, someone would figure out a scam to rip people off if they had your bank account number. I'd feel VERY nervous giving someone my bank account number.
When I lived and ran a business in Asia, most b2b transactions were via direct transfer between bank accounts. Hell, most businesses have their account number right on their website for just such a purpose. People transferred money to friends and relatives all the time, paid for mail/internet order items, paid bills (electric, phone, rent, etc) right at the ATM or over the internet.
Many ATM's accept cash, and you don't even need an ATM card to deposit/send money to someone. You just enter the account number, deposit the cash, and your done.
The US banking system caters to businesses and nickel and dimming consumers with a fee for damn near everything that should come as a free service (incentive) for keeping your money with them.
The idea that anyone would even be afraid of sharing their account number only shows how behind the times the US banking system is.