Thank you all for your comments.
I also believe in trusting our gut’s feelings, although sometimes it’s difficult to know if the feelings are genuine or if perhaps there is some fear in leaving the comfort zone. Either way, it is always good to hear other people’s experiences and thoughts.
Just to clarify some points, the sitting would go ahead if one or both the children would be there.
The little house is separate, the garden where the pet spends most of the time is shared.
The children are grown up, they are College students. I don’t know if the child would have access to the main house or not, it hasn’t been discussed.
I don’t know where the pet would stay, but this is a big one. Also it shades a lot, it requires daily hoovering. If it had access to both houses it would be a massive work to be done.
When during the call I heard that if both children come home I should move to the granny flat, I felt very bad. I’ll be driving 1200+ km to reach their home. I’ll be given the possibility to spend a long time in a Country which I love where I won’t have to worry to pay a rent and I feel grateful for this, but I’ll also provide a service. Staying in an external unit for me, it sounds like being excluded, like being a servant or similar, no matter how beautiful the accommodation is.
Also being asked to move out, no matter how kindly has been said and what are the reasons behind, it just sounds like get out of the way and give space to other people who have the right to stay while you don’t.
The PP very kindly invited me to stay for a few days before and after the sitting. Of course I would be delighted in staying in the small house in these circumstances so that we can give each other’s privacy, but it is a different story from the sitting.
I think it should have been highlighted on the listing. Most of the accommodations I see on listings are in the main house, I’ve seen only very few ones where sitters are accommodated in an external unit. I’m not one of those applying for it, but I know many people are more easy going than I am.
Anyway, I had the answer within me, although I wanted to sleep on it. The day after I sent a message explaining all the reasons and saying that with regret I would decline the offer.
I thought they were going to accept it and that I was going to move on, instead… I got a message where I was told that they totally understand and said that maybe if I could stay in the smaller house since the start, it would make more sense and I wouldn’t need to move out… I thought, they don’t really get it 
Today I got another message where they sent me the video of the smaller house. It was built for the grandparents, despite being smaller than the main house, is quite big, clean, beautiful and all that.
They’re telling me that child n. 2 won’t go home for sure. Child n. 1 could go home and be happy to stay in either house. They’ll know only shortly before the sitting starts. The sitting would go ahead anyway as child will be busy on Summer job.
They’re saying that the other applicants are happy to stay in the small house, but they’re also saying that I’m their first choice.
How flattering it can be, in some way it is getting more complicated… I thought, despite with some regret, that I was this sitting behind my shoulders. Instead, I can see that they’re trying to work it around, but still I wouldn’t feel comfortable.
This is a Country that I know quite well as I lived there for many years and I just “escaped” when the housing crisis started dramatically. People are improvising tiny granny flats in their back gardens and charge tenants very high prices. It has become normal to share space with strangers.
I have my experience in sharing flats when I was younger, but now at my age and on my life path, I’ve become more private. Also in the Country where I’m living now, we have many problems but luckily we all have a home and we don’t need to share renting with other people.