Most people with an intact anal sphincter are not going to lose the small amount of liquid involved in plugging by letting it leak out.
That said if the "pluggee" is not used to this method or using enemas he or she may be more prone to leakage in the upright position than when lying down.
The other part of the lying down suggestion is actually borrowed from recommendations for taking an enema.
It is easier to take a large quantity of fluid when lying down.
If one lies on the left hand side the structure of the bowel is such that the fluid will tend to run into the lower colon, thus easing pressure and the desire to expel.
Hope this helps