Whether you were talking about a specific pair of scissors was related to your original question as it was a very specific example. You would still want to add "a part of" before scissors. "A pair of" is really just like the word "a". Some words need "a pair of" and not "a". A pair of is used with two things that look the same, are the same size and are meant to be used together.
A pair of shoes
A pair of pajamas
A pair of gloves
We also use a pair of for something that is made of two items joined together
A pair of glasses
A pair of scissors
A pair of trousers
Note: A pair of is always followed by a plural noun. E.g. 'A pair of gloves', NOT 'A pair of glove'.
However the sentences you said both could work. It feels like they need something, though. "Can I borrow some scissors?" sounds better. "Can I borrow a pair of scissors?" also works.
"I bought scissors" works, but it could mean that you bought multiple pairs of scissors, so if you want to be clear you only bought a pair of scissors then you would say "I bought a pair of scissors."
Basically use "a pair of" when you are talking about scissors (singular) unless you are talking about a specific pair like in the first comment (using the scissors or those scissors), and you should be fine. If you leave out "a pair of" it could make people think you are talking about scissors (plural). Scissors is a mass noun and they are different than regular nouns. On top of that scissors are one of the handful of words that use this phrase "a pair of" (a pair of scissors/pants/glasses), so scissors is a difficult word sometimes. If you are still confused you should look up mass nouns and nouns that use the phrase "a pair of", and hopefully it will make more sense.
Honestly it's not extremely important as native speakers will understand you meaning even if you make a tiny mistake, which you probably wont if you do as stated above. If you listen to a lot of English eventually you will figure out the difference naturally without even realizing it. Don't worry too much.