What does it feel like to be on the receiving end of it? Is it a tangible thing? How about the practitioner? Do they feel it, too?

Some people become very emotional others feel heat or cold on certain parts of their bodies where the hands of the practioner are "hovering" (he/she normally doesn't touch you)

The 1st time I got a Reiki treatment by my Reiki Master, I felt a sense of well being and relaxation.
If you do not ground yourself before and after the treatment you can remain warm all over - even flushed - for quite a long time which is quite uncomfortable. So, the grounding is also very important.
Some people do not feel anything but the practioner may well be experiencing - on his hands- a cold or wamth feeling while giving you your Reiki treatment.
Also, while you may get a warm feeling on, let's say, your shoulders, the practioner may get a cold feeling while going over your shoulders. As long as the "temperature"remains relatively even on both sides of your shoulders you are OK. It is when you experience a great difference in temperature that a problem can be detected and alleviated either with 1 or more Reiki treatments.
So, yes, I would say it is tangible… I am normally a very sceptical person and I always need a logical explanation for everything but…I am convinced Reiki IS working and playing some magic on our emotional, spiritual and physical health. I am so convinced of the positive effects of Reiki that I became a practioner. I treat friends and family and everybody has felt something while receiving a Reiki treatment. No need to tell you… that I do not practice Reiki for money as some answerers keep on suggesting (and, in doing so, putting all practioners in the same bag) but to try to improve the positive minded attitude and the health of said friends & family.

Hope this helps.
Sending Positive Energy your way!
Love & Light!

Hope this helps.