When I see questions like this, I like to point out that this on-or-at dilemma plays out in other ways, not just with the word side. Both prepositions have a host of meanings, some overlapping, some not. Context is everything!
Sometimes swapping the preposition will cause a sharp change of meaning:
The children were at the table.
The children were on the table.
Please put my slippers on the foot of the bed.
Please put my slippers at the foot of the bed.
(In those examples, on means "atop", while at means "nearby".)
Other times, they can't be swapped; one preposition fits very naturally, while the other creates a seemingly nonsensical expression:
The mouse is on the mousepad.
?-The mouse is at the mousepad.
Stick the stamp on the envelope.
?-Stick the stamp at the envelope.
Dave and Janet are at the movies tonight.
?-Dave and Janet are on the movies tonight.
Dave and Janet should be here at ten o'clock.
?-Dave and Janet should be here on ten o'clock.
Then there are idiomatic uses, where one preposition is more fitting than the other:
During the last minute of the game, the fans were on edge. When
the buzzer sounded, they finally felt at ease. (Or, if the other team won, the fans might have felt at a loss).
But sometimes, the difference between the two is minimal, because one of the several meanings of at (and on) is "in a particular place" (which is why you were having problems with the word "side"):
Draw a small 'X' on the center of the circle.
Draw a small 'X' at the center of the circle.
The fish seem to be biting on that side of the pond.
The fish seem to be biting at that side of the pond.
Which is better in those situations (such as Carter's excellent side of the road examples)? In those cases, it may just boil down to a matter of personal preference. Then again, there could be a clear-cut favorite, depending on the context:
During that difficult time, she was standing at the side of her brother.
During that difficult time, she was standing on the side of her brother.
Here, either on or at could be used, if the sentence is merely saying that she was physically located near her brother's left arm. If the sentence is meant to suggest emotional support, though, then that would be better conveyed with the idiom at his side. However, if there was a feud in progress, and the brother and sister were allies, then on his side would be the best way to express that.
Lastly, you would always use "on the side" is when alluding to a side dish:
At lunch, I ordered a turkey sandwich, with slaw on the side.
I don't think that connected to is an appropriate synonym for became acquainted with: It's too informal and has connotations that cast aspersions on "her" character. I want to ask "How did she become connected to these scholars? Did she sleep with them? Did she marry one of them? Was she hired by one them? Did one of their universities hire her?"
The syntax of this sentence is strange, I think. Rather than
"At the conference, she became acquainted with more than 20 scholars, exhibiting excellent networking skills."
it should probably be
"At the conference, she exhibited excellent networking skills by becoming acquainted with more than twenty scholars."
Tacking "exhibiting excellent networking skills" to the end of the sentence is not good style, IMHO. It modifies nothing. Were it the introductory clause followed by "she became acquainted with... at the conference", then it would modify the following clause. Tacked on to the end of the sentence, though, it just seems ungrammatical to me. I know it's a popular construction for academic writers, because all my Taiwanese biomedical authors constantly use it in sentences that run something like this: "IL-6 levels were higher in group A than in group B, suggesting...." and I always change it to "which suggested that...." Okay, maybe it's just a pet peeve, but it seems to me to be a comma splice.
It'd be different in this sentence: "At the conference, she, exhibiting her excellent networking skills, became acquainted with more than twenty scholars". I still don't like it. The sentence is clunky. It reads poorly to my ears.
Best Answer
The only thing wrong with your assumption is that anyone but a non-native speaker would say "by the bus." As the others have pointed out, the construction would be "by bus" or "by train" or whatever.
Nevertheless, it is certainly possible to say
Here is a movie poster that illustrates what I'm talking about:
Some may object that this might seem ambiguous, that it might seem as if the conference were being held on the bus itself, but in normal informal speech it would be easily understood. More likely the sentence would be reordered in this case, such as