At the same f-stop (1.4), do other 50mm lenses exhibit the same problem? How do the pictures appear with smaller f-stops, such as f8?

In your photo of the couple, it appears that the plane of focus is behind the subjects, but the photo of many people in a line appears sharper on the left than the right. But those photos are just too small and low resolution for analysis. You need to make many more pics to help identify any problem.

Obtain some test charts, or just pages of newspaper with fine printing, place them on a flat surface, and make a series of photos straight-on (at a perpendicular angle). Then publish the entire frame, at a higher resolution, and give detailed information as to the focus distance, the f-stop and shutter used. This will show the extent of the out of focus areas, and whether it is consistent across the flat plane.

Then place a ruler or yardstick in front of the lens, pointing away from the lens, and focus on the mid-point. Make a series of photos at various f-stops to determine the depth of focus.

Finally, try stopping the lens down prior to focus. You should be able to focus stopped down in bright conditions to around f5.6.

I doubt there is anything wrong with the lens. At f1.2 the lens will normally be difficult to get exact focus with any camera, but at smaller apertures it should focus the same as your 50mm f1.4. But with more test pics, a mis-aligned camera focus screen, or an out of square camera lens mount can be identified. And if you do comparison shots, at the same aperture and distance, with the 50mm f1.2 and your 50mm f1.4, any problems with that specific f1.2 lens can be easier identified. Shoot some more test pics, and show us the results.