Grammatically correct: In this place there are many people with whom I can speak.
Colloquially, we say: In this place there are many people who I can speak with.
Technically, it should be "whom", especially with the original sentence structure. However, we usually just say "who" in every day speech, but please note that when doing so, the structure of the sentence changes and "with" comes at the end.
With whom are you going? (correct)
Who are you going with? (grammatically incorrect, but sounds natural, very common)
With who are you going? (incorrect and sounds weird. no one ever says it this way)
Whom are you going with? (arguably grammatically correct, sounds OK... uncommon)
By not separating the placement of the "with" and "who" in the sentence, it makes it very apparent that "who" is wrong in the phrase "with who". (Hope this isn't too confusing of an explanation.)