Rear Cassette Cog/Sprocket is loose

I replaced my chain on the weekend and was cleaning my bike and noticed when fitting the new chain that the top Sprocket/Cog was loose and wobbly. I noticed that this was riveted, as I thought it was screwed in. Is there a way to replace this without purchasing a whole new derailleur. I'm not sure what type to get as it's a Deore Shadow Plus 10 speed.
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If so, it is held in place with a bolt so easy enough to replace.
The bolt head for the bottom jockey wheel is on the outside of the cage but the upper one is on the inside of the cage.
Bear in mind that upper jockey wheel will have some play in it as it acts as a guide.
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Google Shimano Jockey wheels 10-speed
Like JBA says a few mm's of lateral float is perfectly normal to aid smooth shifting and fine tuning but there shouldn't be any sloppiness or twisting movement.
Or maybe the two on the left instead of the two new ones on the right?
Then they are worn out and it's time to change. Jockey wheels can be badly worn and still function, but you will notice crisper shifting once you fit new ones. If the bearings are totally shot, then there is a risk that they will disintegrate on the trail, and leave you stranded. And you don't want that!