Khaleesi of Nothing, the Millionth of your Name
Each of the old women had been a khaleesi once. When their lord husbands died and a new khal took his place at the front of his riders, with a new khaleesi mounted beside him, they were sent here, to reign over the vast Dothraki nation. Even the mightiest of khals bowed to the wisdom and authority of the dosh khaleen. Still, it gave Dany the shivers to think that one day she might be sent to join them, whether she willed it or no. (AGOT, Daenerys V)
The heart of a stallion would make her son strong and swift and fearless, or so the Dothraki believed, but only if the mother could eat it all. If she choked on the blood or retched up the flesh, the omens were less favorable; the child might be stillborn, or come forth weak, deformed, or female. (AGOT, Daenerys V)
"You belong to the Dothraki now. In your womb rides the stallion who mounts the world." He held out his cup, and a slave filled it with fermented mare's milk, sour-smelling and thick with clots.
Dany waved her away. Even the smell of it made her feel ill, and she would take no chances of bringing up the horse heart she had forced herself to eat.
Do the dosh khaleen have power?
They certainly have status and influence. They have important jobs in the temple. They are probably kept safe. Compared to most of the other options, it’s nice work if you can get it.
But it’s also not something that you can aspire to, earn, or even choose. When Daenerys first sees the city as a teenager, she is adapting to a situation that’s been forced onto her. While she intellectually acknowledges that the temple of the dosh khaleen is how that situation ends, she doesn’t come up against what that means until Drogo is wounded. Before anyone knows about his injury, she can pretty much do whatever she wants, as long as she explicitly invokes his authority. Even after the bloodriders realize it’s serious, he’s still alive and conscious enough that she can convince him to submit to Mirri’s treatment. Her idea is pretty transgressive and it’s impressive that she fights for it – but even then, the issue is whether Mirri’s abilities can be exercised, not whether Dany can do anything on her own. But she will lose everything with his death, and she knows it. Much as she’s come to care for him, her fear of his death is largely and explicitly about how she will lose everything when the light of her sun-and-stars goes out. And she’s right!
Jhogo took the whip from her hands, but his face was confused. "Khaleesi," he said hesitantly, "this is not done. It would shame me, to be bloodrider to a woman.".... Aggo accepted the bow with lowered eyes. "I cannot say these words. Only a man can lead a khalasar or name a ko."..... "You are khaleesi," Rakharo said, taking the arakh. "I shall ride at your side to Vaes Dothrak beneath the Mother of Mountains, and keep you safe from harm until you take your place with the crones of the dosh khaleen. No more can I promise."
A khaleesi has status, and some derivative degree of influence. But it’s not a title which connotes authority or power, individually or collectively.
That’s not to say they’re irrelevant. The dosh khaleen serves an important purpose in Dothraki life. They are the inhabitants and keepers of Vaes Dothrak. The city is both domestic and public, effectively serving the purpose of the private sphere for the Dothraki: it provides a place of sanctuary where the men who run a society are protected from competition with other men. But it also serves the public function of allowing the various khals to maximize the benefits of cooperation and trade while minimizing the risks. The dual nature of the city creates the sense of it as a home for all Dothraki, which in turn supports a sense of shared community across the many far-flung khalasars. Vaes Dothrak is important enough that the constraints on its inhabitants are critical.