Networking in London

I’m an Associate 0 in London, looking to make a move to PE. Ive been in touch with HHs but want to take my search into my own hands as well. My question is - how is networking received in London? Will investment professionals at funds be receptive to a cold email / LinkedIn message asking for a call? Is this ever done?

Additionally I am a bit concerned about it getting back to my current Bank. My plan would be to target those with the same background as me, which obviously leads to there being some mutual connections. Is there not a very high chance that this would ultimately get back to my current team in that case? That would obviously not be a good look!

Many thanks!