Generally beam dispersions are specified as the half-angle to which the intensity is half that at the center. Photometrists assume an ideal beam which is round and smooth, none of those Maglite rings, Luxeon-5-watter donuts and Surefire ovals. Angle measures start in the middle and go outward since they see little need to measure the same thing twice - remember they assume the spot is the same all the way round. The full-angle does get quoted now and then though, that`s just twice the half angle - from the half-intensity point on one side going through the center spot to the half-intensity point on the other side.
I use a beam target calibrated to give half-angle values when the light is 50cm away form the surface, which is fine for small lights like LEDs but does give false results with tighter beams such as Spotlights and similar long throwing incandescents - the beam isn`t fully formed at such close distances. I just eyeball the approximate half-angle from the digital photo of each beam profile, but a more scientific way would be to use a light meter with a small photocell (not one of those large-window ones), measure the brightness in the middle then slowly move out till it reads half that peak value. That`s your half-angle right there. You don`t need a target to do this though - as Don suggests, laying the light on a flat surface with the beam shining along will also allow you to measure its angle. If you remember Trigonometry and stuff like that from school (I don`t!) you can measure the intensity and calculate the angle at whatever distance you choose, space permitting of course.
Bean angles on the other hand, can be measured by placing the bean on top of a protractor with the curved part of the bean at the middle zero-point. I can draw a diagram if it helps. Light meters don`t work too well at determining their half-angles.....
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