When I paint in vertex paint, and after I switch back to object mode it does not show the paint This is the situation here
https://i.gyazo.com/79225d6fd878431834b8b27ea54f59b0.mp4

As you can see when im in vertex paint I can see what I have painted, but once I go into object mode I can no longer see it.
To see the vertex paint in Object mode you must switch the display mode to Texture.

See the image below showing the vertex showing in Object mode with Texture active: