How do check valves regulate the air in a multi-engine airplane's vacuum system?
On each side of the check valve manifold there is a check valve that only allows air to flow toward the vacuum pump on that side.  For the left pump, the check valve is the rectangular bit between the far left line going to the vacuum gauge and the adjacent line going to the deflate valve, and for the right pump it is between the far right line and the gyro suction line.

The result is if one vacuum pump no longer is producing suction, the vacuum gauge indication on that side, being "upstream" of the check valve now that only the opposite pump is pulling air, will show no vacuum; meanwhile the operating pump can still pull air through the other three lines and can still operate the deflate valve and vacuum gyros.  Its vacuum regulator will control the vacuum level.  Everything will continue to work normally with only the loss of vacuum indication for the failed pump providing an indication that something is wrong.