If you are seeing your vehicle as having passed the smog check when you search BAR Smog Check History
results, then you can be confident the smog station did in fact electroincally send your smog check results to the DMV. The DMV should not require a "hard copy" of the smog certificate. For added surety you may want to print out the webpage from the BAR Smog Check History search. For vehicle registration and title transfer purposes, the DMV in most circumstances, will not require a hard-copy of your vehicle's smog check results, smog certificate or Vehicle Inspection report (VIR).
As for getting a "hard copy"... A copy of your vehicle's smog certificate (also known as the Vehicle Inspection Report) can be obtained only from the smog station which inspected your vehicle; and might require a service fee. In most instances the smog testing center will provide a duplicate free of charge. Contact the smog station where your vehicle was inspected.
Smog certificate information is kept in electronic format in the VID (Vehicle Information Database) which is run by the State of California Bureau of Automotive Repairs, and looked over by the California Air Resources Board. Once a vehicle receives an official smog check, the smog inspection data, including pass or fail results, are stored with the State and available to government entities, such as the California DMV, indefinitely.