1. Telepathy is a HOT topic, especially now that so many have commented on it.
2. Every person has an opinion.
3. Every person has a different mental collection of perceptions. Individuality is secured in those personal conceptions. You ARE the proof, Susan.
4. Silence in one's mind conveys ineffable proofs of a subjective kind; unique to that person. Yet, every person is so likewise endowed with the same attributes of mind and body.
5. Simply put, a "Common" universe of space and time is the cherished materialism that all people share together. Any differences are of thought and behavior, which "moves" physicality ever so little. Our words do we communicate our individual mental "universe". Absolutes that come upon our own perceptions are, indeed, conjecture. Yet, every person who lives is attached to a variation of the "common" universe. Thereby, our own "directives" of fact and proof are indeed subjective, and also influential to others.
6. Now, since this OP is about telepathy, the explanation of only one person is a fair and just "assessment" of the "common" universe. I have "visitors" to my perceptions that come to me by an unspoken agreement to "allow" access to my mind. This is "controlled telepathy", just as Gerda has explained. "Uncontrolled" telepathy is what Dion Fortune has coined as a "psychic attack". One must always be on guard that no other person attempt to control one's personal consciousness. These are the unwelcome strangers that interfere with one's own thought-flow. "Strangers" may also be specific "world-ideas", like Nazism, narcisism, hate and other inhibiting ideas brought in from the "world-arena".
7. Your use of the word "independent" observations when doing an experiment assume that every fallacy of inconclusive experimentation is not proof of anything. It is the "half-empty" view, not the "half-full" view. No experimentation is absolutely flawless. Yet, the outcomes must indicate a measurable pattern of repeatable procedures. This is both How and Why science is "wallowing" in the mire of the "Uncertainty principle".