Vivian Nutton provides a detailed account for the workings for a Greek/Roman temple like the Asclepeion in his book Ancient Medicine pp. 109-110.
At the shrine suppliants would purify themselves at a sacred spring,
before offering an appropriate sacrifice, and then, wearing white
robes, undergo a second purification before entering the abaton or an
adyton, ‘the inaccessible’, words that stress that it is a building
barred to the normal visitor. Only those prepared to meet the god or
to serve him as a priest were allowed to enter or to find out what
actually took place within. A man called Aeschines, who climbed a tree
to see if he could see what was happening when the suppliants were
asleep, was punished by falling on to a fence and nearly losing his
sight. The abaton itself was a long porticoed edifice with distinctive
individual rooms: when no such building existed, as in the early years
at Athens, it was enough to sleep within the temple itself or perhaps
even its precinct. If the suppliants were fortunate, while asleep they
would receive a vision from Asclepius. In it sometimes the god himself
appeared and healed them by acting as a physician or surgeon;
sometimes it was one of the sacred snakes or dogs who appeared to lick
or enter the person; sometimes the dream itself was a mere riddle and
required further assistance to be understood. On waking, the sufferer
might be completely recovered, all paralysis or swellings gone, but
sometimes the god had given instructions which needed to be
interpreted by a priest or temple guardian and then followed up before
a cure was secured. Many of the treatments find parallels within
contemporary medicine, but others were perhaps selected for public
display precisely because of their striking divergences from it. But
to think of the healing encounter solely in terms of medical
techniques is to miss the context in which it takes place – the
physical setting, the sacred spring, the sacred grove (even if, as at
the Asclepieion at Athens, it could have hardly amounted to more than
three or four trees), the sacrifices, and the reassurance offered by
the memorials, whether inscriptions or cultic recitations, that this
was a place where healing was available.
Source: Ancient Medicine (2004), pp.109-110.