In Hard to Be a God, the main character (Anton/Rumanta) falls in love with Kira and decides to take her with him back to Earth.
Since their task is to be as undercover as possible, it’s obvious that they are not allowed to save everyone/whoever they like. Kira also has not any bigger roles to build up Arkanar after the revolt.
How do they decide whom to save/what are the rules on who is allowed to bring to Earth?
They don't bring anyone to Earth.
The persons of interest — scientists, astronomers, medics, artisans, poets, you name it — are not sent to Earth. They are re-routed to kingdoms that value their respective fields more than Arkanar (which is pretty much any kingdom out there).
Bagheer of Kissen, accused of lunacy bordering on treason, had been thrown in a dungeon and had only been rescued by Rumata with great difficulty. He had been sent to the metropole
[...]
He had then sent Sinda, who hadn’t understood a thing, into Soan, where he shrugged his shoulders and continued to look for the philosopher’s stone under the watchful eye of Don Condor.
Hard to be a God, translated by Olena Bormashenko, printed by Gollancz UK first edition, chapter 5, page 117, all bold emphasis here and hereafter is mine.
As seen on this page in the book, Rumata (and, I reason, other operatives) only target people who would immediately be neutralised without operatives' intervention. The whole point here is doing as few interventions as possible, and letting the history take its course. Operatives are mostly observers, and the action part comes later.
“I don’t like that we've tied our hands and feet with the very formulation of the problem. I don’t like that it’s called the Problem of Nonviolent Impact. Because under my conditions, that means a scientifically justified inaction."
[...]
“There are two hundred and fifty of us on this entire planet. Everybody controls themselves, and everybody finds it very hard. The most experienced of us have lived here for twenty-two years. They came here as nothing more than observers. They were completely forbidden to do anything whatsoever. Imagine that for a moment: forbidden to do anything. They wouldn't even have had the right to save Budach. Even if Budach was being trampled before their eyes.”
Ibid, chapter 1, pp. 37-38.
There is no indication Anton would be allowed to bring Kira to Earth.
There are two conversations about Kira's possible travel to Earth, both of which seem more like Anton's daydreaming, than actual plans. It is, in my opinion, highly unlikely that Anton would be allowed to bring Kira anywhere, and there is no mention of other operatives keeping souvenirs either.
There is this one passage where he thinks whether he could ask his friend Anka to keep an eye on Kira when she goes to Earth:
You’d be safest on Earth, he thought. But how would you manage without me? And how will I manage here alone? I could ask Anka to be a friend to you there. But how will I manage here without you? No, we will fly to Earth together. I’ll pilot the ship myself, and you will sit next to me, and I’ll explain everything to you.
Ibid, chapter 10, pp. 225-226.
... Although Boris Strugatsky thinks otherwise.
Boris Strugatsky has been doing an off-line interview with fans since 1998. Among the questions about Hard to be a God, someone asked whether Rumata would actually be able to bring Kira to Earth. The answer was, "Why not?" (see question 10):
B: Действительно ли Румата мог взять Киру и улететь на Землю?
O: Мог бы. А почему, собственно, нет?
Q: Is it true that Rumata was able to fly to Earth and take Kira with him?
A: He could. Why not?
Translation mine.
If Anton was allowed to bring Kira home, it would be precisely because she is insignificant.
Who is Kira? No one! She is a daughter of an assistant to a court clerk and a gray sergeant. Not married and a redhead.
There was nothing extraordinary about her. She was just a girl, eighteen years of age, snub-nosed, her father an assistant to the court clerk, her brother a sergeant in the storm troopers. And she was late getting married, because she was a redhead, and Arkanar didn't think much of redheads.
Ibid, chapter 3, page 79.
Fact: her father is a literate man. Chances are, he'll get caught in the whirlwind of Don Reba's purge and be slaughtered, especially after
The Holy Order takes over Arkanar.
The same goes for her brother. Due to her red hair, there's a fair chance she'd be killed too, for the reason above.
... and this is almost what happened, as we see in later chapters:
Her father was laid up. He had been kicked out of his office and beaten severely with sticks as a farewell. [...] The boy also said that her brother had turned up—wounded, but cheerful and drunk, in a new uniform. He gave money to his father, drank with him, and was once again threatening that his boys would roll over everyone. [...] Father asked that she not come home under any circumstances. Her brother was threatening to settle scores with her for getting mixed up with a noble, the red-haired bitch.
Ibid, chapter 10, pp. 224-225.
