Laser Ranging Telescope.
http://stardate.org/images/gallery/iotw_20090717.jpg

No, it is not possible to detect the beam on the moon by sight, but there is a device on Earth that verifies the laser signal returning from the moon.

There is a laser device in the McDonald Observatory Laser Ranging Station near Fort Davis, Texas, which regularly sends a laser beam through an optical telescope to try to hit a reflector left in the moon by the Apollo 11 mission in 1969. The reflector is a mosaic of 100 glass half-cubes, an experiment left on the lunar surface 40 years ago which still continues to return valuable data about our satellite.

This reflector requires no power and is still functioning perfectly. It is far too small to be seen from Earth. Even when the beam is correctly aligned in the telescope, actually hitting a lunar reflector is quite challenging. At the Moon's surface the laser beam is a little over a kilometre-and-a-half wide! The reflected light is too weak to be seen with the human eye.

The pulses of laser light fired from the Laser Ranging Station on Earth are reflected by the lunar laser ranging reflector and a tiny bit of its light will return to the telescope. The idea was to determine the round-trip travel time of a laser pulse from the Earth to the Moon and back again, thereby calculating the distance between the two with incredible accuracy.

The data gathered has shown that the Moon is receding from Earth at about 3.8 centimetres (1.5 inches) every year. Scientists now know the average distance between the centres of the Earth and the Moon is 385,000 kilometres (239,000 miles). This level of accuracy represents one of the most precise distance measurements ever made.

McDonald Observatory Laser Ranging Station.
http://theprobs.com/wp-content/uploads/2009/08/Figure-11.JPG

Apollo 11 mission ranging reflector.
http://history.nasa.gov/alsj/a11LRRR5952.jpg
Source(s):
http://news.bbc.co.uk/2/hi/science/nature/399468.stm

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