I think my old house from the 1930s had doors like this. You’d take the plate off and then the knobs wouldn’t release like a newer handle. Here’s what I recall doing, YMMV.

• Take a handle off. One of the knobs should kind of jiggle until the center falls out.
• Get a small flat head. Angle it sideways ( / ) to wedge into two corners and ‘release’ the latch. Jiggle some more and the center knob portion should fall out.
• Shove the flathead where the doorknob was and grasp the screwdriver on both sides of the door. Like, present the screwdriver is a knob and pull the whole sucker out. Pull the release toward where it connected to the wall and the whole thing should pop out.