Yes a photo would be good to ascertain what exactly your after
Is it important that the mortar joints be also flush on the back side in a double brick cavity construction house.
It seems common practice these days that the mortar does not completly fill the back side bed joints whereas in the old times the brickies would have to scrape the excess from both sides of the bricks.
If normal practice what is an acceptable depth of void?
In the old times the bricks were pressed with a frog which spewed the mortar out easier.
Not having mortar spew out the back means its not going down the cavity which is more important. Really we need to see the type of bricks you discuss.
A lot of people complain about no mortar in the perpends in internal bricks except that many bricks now such as the fastwall are designed that the vertical perpend joints are left open so that the render/plaster sticks to the wall.
Some builders have told brickies to fill the perpends to appease owners so the job looks better however it is very self defeating of the original brick design intention that the perpend joints be left open to take the render so it doesn't fall off the wall.
How wide are the bricks? You have a photo. Your better off not having mortar falling down or bridging the cavity leaving the structure open to water ingress across the cavity.
Cavity walls are designed so that rain hits the outside wall then runs down the back of the outside wall and out at the base through weep holes "without" crossing to the internal wall causing internal dampness via capillary action leaving damp patches.
Currently obsessed with non-hydraulic mortars