Hyphenation of "left hand side"

Do whatever you please; all three of

1. right hand
2. right-hand
3. righthand

are readily encountered in the wild. It’s possible that right-hand is the variant currently the most used, but your environment may not share this predilection.

But I have a better suggestion: omit hand altogether.

The word "righthand" is perhaps used somewhere, but it is not recognized by the Oxford Dictionary 12th edition Concise. It's up to you which authoritative reference you wish to adhere to. The important thing is to be consistent throughout, so your readers will know what terms you are using and why. If you stick to the Oxford Dictionary as reference, then the use of a hyphen simply follows the rule of attributive (before the noun) or predictive (after the noun) adjectives. That is to say:

The right-hand side of the entrance. The wall on the right hand. In this sense, I guess most of your usage would be with the hyphen.

Confusion or ambiguity should be eliminated by use of additional phrases: (the right-hand side of the entrance as faced from the outside by the observer..........or something less clumsy - for you to reflect over).

But, finally, wouldn't it be simpler just to use "right side" and "left side" "On the right side of the entrance" ?