Using QGIS 3.4.3, the shape files created don't fit with Bing aerial plugin only on the right side of the screen, on the other hand, for the same screen, the left side looks ok. When I move the non coincident point (for instance a cross road digitized shape and the same point on bing aerial on background)located at the right side of the screen to the left of the screen, it becomes coincident. Can anyone explain me why this is happening and how to avoid that?
This frequently happens with data reprojected on the fly, especially rasters reprojected on the fly. If you download a copy of the raster you want and reproject it using the raster reprojection command, the problem should disappear.
Your question says QGIS 3.4.3, but the screenshot clearly shows that your QGIS version is 2.12.3.
If you're using QGIS 2.12, stop using the OpenLayers plugin. This plugin was extremely buggy in earlier versions of QGIS. In addition to repeatedly changing your project CRS to EPSG:3857, it often loaded layers in the wrong location. Use the QuickMapServices plugin instead.