Sit beside me./ Sit down beside me.
Is there any difference in perception of these two phrases by people? (I am interested in how the word "down" changes the meaning of the phrase.)
If I were a native speaker of English, I would never shut up. :-)
There is no appreciable difference between them. In this case, the word "down" makes no difference. We use "sit" and "sit down" interchangeably in many situations.
Sit down here.
Sit over there.
Sit down over there.
Having said that, they are only the same when we are using them for the action of going from a standing to a seated position. "Sit down" can't be used in something like "I used to sit next to my best friend in maths".
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.