Q: Is "on the side" an adverbial phrase you can plug anywhere? I am wondering if you can plug "on the side" (adjacent to me) as an adverbial phrase you can plug anywhere.

She stood next to me when I was watching the TV and stared at me on
  the side.

Can you think of an example where you can't plug "on the side" after a verb + object group?

A: No, you cannot just use that phrase anywhere. The construction is syntactically correct, but it has semantic issues.
In your sentence, on the side doesn't make sense. What are you trying to describe?


*

*She was standing beside you and looking at you.  

*She was staring at your side.  

*She was looking at you sideways.

*Something else.


Typically, on the side has a few meanings:


*

*In addition to. I'd like a sandwich with some fries on the side.  

*A figurative term for cheating. He was seeing someone else on the side.  

*A literal term. I was standing on the side of the crowd.


But stared at me on the side doesn't correspond to any of the normal meanings, so it sounds strange. Whatever it is that is meant would likely not be understood. Somebody hearing that sentence would probably ask, "What do you mean?"
