The company/person who issues the paycheque is responsible for issuing the paystubs too, because the company also has to keep a record of your paystub. If you get your paystub electronically deposited, the paystub may be given to you the next day by a manager, or maybe it's mailed. Depends on company policy.

When I say company/person, I mean the one who actually does the payroll (usually someone in a payroll department). A lot of companies now are using payroll services, and they are the ones that acutally prints the cheques or directly deposits the amounts to bank accounts. But whoever does the actual payroll payment does the cheque stubs too.

The manager is usually just responsible for giving the total hours worked for each week and the rate the hours are paid. She (or he) would then give the stubs to the employees when she receives them from whoever does payroll. She proabably would also have to give acutal paycheques to anyone requesting that their paycheques not be direct deposited (it happens - out of about 40 employees, I used to have about 5 who wanted an actual cheque).