We don't know exactly what piece of machinery did what in this process.  In "Trouble with Tribbles", we get another example of people beaming  from one Federation pad to another, and there, too, it's unclear whether  the 
Enterprise machinery or the K-7 machinery was doing the  work, or whether both were involved. In contrast, in "Ultimate Computer"  we see Scotty operating the sliders when people beam aboard from the  local space station...
The dialogue in ST:TMP is ambiguous enough.
	
	
		
		
			Cleary: "Transporter room, come in! Urgent! Redline on the transporters, Mister Scott!"
 Scotty: "Transporter room, do not engage! Do not-"
Cleary: "It's too late. They're beaming now!"
Rand: "Do you read me, Starfleet? Override us. Pull them back!"
 Kirk: "Give it to me. Starfleet, boost your matter gain, we need more signal! ...More signal!"
Scotty: "We're losing their pattern."
Rand: "Oh, no! They're forming!"
Kirk: "Starfleet, do you have them?"
Starfleet: "Enterprise... What we got back didn't live long... Fortunately."
		
		
	 
What  we learn is that Rand "engaged", which is not necessarily the same as  her energizing the starship's machinery. Might be her simply signaling  the go-ahead for Starfleet to energize. And it's not clear who Cleary is  referring to when saying "They" are beaming - is it the transporter  room, is it Starfleet, or is it the transportees?
Starfleet  is the party capable of overriding and then boosting matter gain and  signal. Does this mean 
Enterprise was initially doing the matter  gain and signal thing, and Starfleet overrode that? Did Starfleet ever manage to override anything?
"Matter  gain" is unique technobabble never encountered again, but "signal" is  something that in later Trek is part of the overall process and not  necessarily pointing to any particular sending or receiving end.
It's possible, then, that
1) 
Enterprise did all the beaming, and the hiccuping machinery made a mess of it
2) 
Enterprise did the beaming until Starfleet overrode
3)  
Enterprise and Starfleet did a joint beaming (although those are  a rare thing elsewhere in Trek, and considered a special boosting  technique for demanding conditions in TNG "Realm of Fear")
4)  Starfleet did all the beaming (although if the 
Enterprise  platform was merely an inert target, then it's difficult to see how  
Enterprise machinery problems would affect the beaming.)
Even  #4 is a distinct possibility, as it may be there was nothing wrong with  the transporters at either end. Rather, the problem aboard the starship  may have resulted in shields being half-raised, or some dangerous  source of radiation being activated, with the end result that even fully  functional transporters would kill their customers, unless signal were boosted.
Timo Saloniemi