Pain is normally thought to be a universal experience, but modern theories like the gate control theory of pain challenge that, as do traditional Yogi practices and other spiritual traditions that focus on simply "not feeling pain".
The Brahmin of Hinduism focus strictly on the experience of breathing as the most basic experience of any person. This was expanded into a broader philosophy of meditation and limiting experience that is allowed to be controlled or constrained from outside, notably in Buddhism.
An early belief of some philosophers of Ancient Greece was that the mind was like a recording device and simply kept somehow-objective records of what the senses experienced. This was believed in the Western world into the 20th century until cognitive psychology experiments decisively proved that it was not true, and that many events were simply filled in by the mind, based on what "should be". This among other things explained why eyewitness accounts of events often were so widely varied.
Another belief, still somewhat more credible, and associated with Ancient Rome, was that personal experience was part of some divine or species-wide collective experience. This gave rise to notions of racial memory, national mission, and such notions as racism and patriotism. It was likely easier to create political movements and military morale with such notions, than a strictly personal idea of experience. Carl Jung and Joseph Campbell were notable investigators of these ideas of collective experience in the 20th century.
During The Enlightenment, there was rigorous investigation of these ideas. Immanuel Kant for instance noted that it was only possible to explain "experience and its objects" as a consequence of each other: Either experience makes those objects possible, or those objects make experience possible. This is seen today as dualism, and denying the possibility of a third thing making both experience and whatever reality its objects have, both possible. That thing could be a more universal cognition, as proposed in some versions of Christianity or Gaia philosophy.
However, by far the most cogent understanding of shared personal experience is in literature, notably the diary and autobiography forms, where an author relates their personal experience in natural language to another. In the Western world, this form is often credited to Augustine's Confessions as the earliest (4th century) example.
Poetry, verse, dance, song and other more abstract forms are thought to relate personal experience better than prose or play by some. It is not clear why this is, other than, the more abstract forms have no one explicit meaning, and experience is more negotiable between artist and audience. These works are often prized in Western literature above works of non-fiction that pretend to be "objective" or about something other than the author. Postmodernism is a movement that denies that anything is objective.
The elements of universal personal experience is today explored rigorously in the cognitive science of mathematics which proposes that our notations and symbols, even the most formal, are rooted ultimately in the senses and habits of exploring our surroundings. Knowledge then strictly relies on experience, and is translated into symbols in an obvious way. Truth is approached by transcription. Baudrillard had a similar theory but did not relate it directly to mathematics.
Subjective personal experience is also more widely spread on the Internet due to webcam and other means. Steve Mann is a noted explorer of ways to share personal experience using the Internet - he makes his audiovisual experience sharable by wearing cameras and microphones literally all the time.