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From Longman dictionary, We know that:
go out to: (move abroad) to travel to another country in order to live and work there. example: They are looking for nurses to go out to Saudi Arabia.
I am not sure about the meaning of the above example. The nurses move to Saudi Arabia, or "They" move to Saudi Arabia and want to get Nurses job.
Consider the situation that they (whoever they are) are looking for nurses in their current location, for example the UK, to go elsewhere (to Saudi Arabia) to work.
Neelam's answer is close, but this sentence implies that they might not go with the nurses, but instead that the nurses go without them, perhaps on their behalf.
For example, in a conversation it could be:
"My boss is looking for nurses to go out to Saudi Arabia, would you perhaps be interested?"
So this sentence means that they (a group of people, maybe a company) want nurses to go work in Saudi Arabia for them.
For context, I'm a native Australian English speaker.
Grammatical analyses (object, complement, modifier) aside, go out to {some place} reflects the speaker's attitude that the place is remote.
A speaker in New York City might say:
They are going out to Alaska to live closer to Nature.
We are going out to the country for the weekend.
Remoteness is relative.
Your first interpretation is correct: the nurses are going to Saudi Arabia.
Remember that English word order is usually just Subject, Verb, Object. The subject of this sentence is They, and the verb (or verb phrase) is are...and what are they? Looking for something. And the object that they're looking for is nurses to go out to Saudi Arabia. So the object of this sentence is nurses: they are looking for nurses, and to go out to Saudi Arabia is a prepositional phrase that explains something about the object.
If the sentence had the other meaning - that they want to go to Saudi Arabia and get jobs - it would have to be phrased like
"They are looking to go out to Saudi Arabia for nursing jobs" or
"They are looking to go out to Saudi Arabia to get nursing jobs" or
"They are looking to go out to Saudi Arabia to become nurses" or maybe
"They are looking to go out to Saudi Arabia as nurses"
But never "they are looking for nurses". Note that whatever they are looking for always comes right after the words looking for.
Your problem is the word "they". There is no particular group of people who are "they" - it's just the way that English says "unspecified people are looking for" in the active voice.
In the passive voice, it's clearer. "Nurses are being looked for to go to Saudi Arabia." It's clear that there is no particular person doing the looking.
Some languages have a word which covers this in the active voice. In French for instance, we would say "on cherche". English used to have this too, with the word "one" - it used to be possible to say "one is looking for nurses". This has long since fallen out of use though.
