How to decide on which side of the runway the approach path indicator (VASI, PAPI) are placed?
Path/slope visual indicators are usually located on the left side of the runway (as seen when approaching), but may be on the right side when impractical. Good reasons for moving the aid on the right side:


Obstacle clearance, terrain, intersecting runways or taxiways.
Interference may occur on the left side, between PAPI lights and nearby lights. This is emphasized in FAA AC 150/5340-30G - Design and Installation Details for Airport Visual Aids: "Because PAPI system is dependent upon the pilot seeing a red and/or white signal from the light units, care should be taken to assure that no other lights are located close enough to the system to interfere with the signal presentation."


Installing them on both sides is beyond ICAO recommendations and FAA orders, but can be done:


For visibility improvement
In order to give better roll guidance


Some countries don't implement ICAO rules the same way, for instance in the UK, PAPI for governmental airports "may be provided on both sides of the runway" (source)

For airports providing a visual guidance for STOL operations. The PAPI associated with the steeper approach path may be located on the right side of the runway (source).


PAPI at the usual location, on the left side of the runway (source).

From Wikipedia:


  The international standard for PAPI is published by the International
  Civil Aviation Organisation (ICAO) in Aerodromes, Annex 14 to the
  Convention on International Civil Aviation, Volume 1, Chapter 5.
  National regulations generally adopt the standards and recommended
  practices published by ICAO.


From FAA JO 6850.2B - Visual Guidance Lighting Systems:


  The basic PAPI configuration is described in chapter 1, section 2,
  paragraph 16, and is shown in figure 5-1. All lamphousing assemblies
  (LHA's) should be located on the left side of the runway, as viewed
  from the approach direction. However, where terrain, cross runways,
  or taxiways make this arrangement impractical, the light units may
  be located on the right side of the runway.
  
  [...]
  
  The VISUAL APPROACH SLOPE INDICATOR (VASI) system is no longer 
  installed by the FAA as a new facility. Specific details and installation
  criteria have been moved to Appendix E to provide technical support for
  existing VASI equipment and facilities.
  
  [...]
  
  The basic VASI configurations are described in appendix D, section
  1, paragraph 1 and shown in figures 1, 2, 4, and 5. All boxes (or 
  light units) of the two-box; four-box, and six-box configurations are
  located on the left side of the runway (as viewed from the approach 
  direction). However, where terrain or cross runways, etc., make this
  impractical, they may be located on the right side of the runway.


Washington Dulles Int'l Airport  offer several location variations, chosen for practical reasons. There is a PAPI on the left side for runway 1C and on the right side for 19C, but a reverse scheme is used for 1R/19L, and 1L/19R is just standard: