How would noble families go about arranging marriages? Did they just out and out say it, or was there more subtly and manipulation? How did they breach the subject and negotiate it?
If the families were well-known to each other and their interests likely to align – the most common scenario we’re talking about is neighbors looking to consolidate their estates – you’d probably be pretty frank about it.
If on the other hand, the two families are not well-known to one another, they’d probably get more subtle about it: have the two sides meet at social occasions, see how the young people in question get along (this is why formal balls and dances were created, and why it’s very strange that cotillion culture still exists), then bring it up in such a way that neither side could lose face, probably as a hypothetical or something.
On the other hand, we have plenty of examples in Westeros of people being very blunt about offers out of the blue and rejected offers leading to hurt feelings, so…
There are, in my opinion, two very good examples of this sort of probing and trusting and manipulation in the text: Alys Karstark and Jaime Lannister.
Lord Rickard was very blunt about what he was doing when laying it out to his daughter, but not so frank and open with Eddard; Rickard was, ostensibly and publicly, going to Winterfell to have a tete-a-tete with his liege lord, like you do, but in reality is instructing his daughter to “charm” said liege lords heir, obviously with an eye to that heir viewing her as a potential friend or romantic partner. And then you can approach Lord Eddard and say “hey, our kids are getting along great, there’s clearly a spark there, we can make them both happy and also maybe once again show how close our houses are.” But he’s very much trying to manipulate the Stark’s here without telling them what he really wants is a marriage contract.
Conversely, you have Jaime Lannister and Lysa Tully. The Tullys and the Lannisters were in talks to see about affiancing them, which were tolerably far along. But Hoster didn’t tell this to Lysa, nor Tywin to Jaime. Instead Jaime was sent to Riverrun with an urgent message that “couldn’t be entrusted to a raven.” And he’s told to wait for Lord Hoster’s response, which of course Hoster spends some days considering. Which means Jaime is at table with and socially exposed to his unbeknownst-to-him possible wife Lysa every night. The kids don’t know what’s happening here, but the adults do.
You’ve also got Oberyn and Elia’s Grand Tour around the Reach and the Westerlands to scope out potential matches for the both of them, where they were aware of some of what was happening and ignorant of other aspects.
Excellent examples!
Just to throw in some counter-examples:
Maege Mormont basically throwing her daughters at Robb, and Lord Hornwood being a bit more nudge-nudge but still clearly trying to hard-sell Robb now that dad’s not around and the kid’s in charge for the first time.
Tywin and Hoster, Round 2: “When I suggested to Lord Hoster that Lysa might be wed to you instead, he replied that he wanted a whole man for his daughter.“
Tywin and Aerys II: “Lord Tywin chose that very night to suggest that it was past time the king’s heir wed and produced an heir of his own; he proposed his own daughter, Cersei, as wife for the crown prince. Aerys II rejected this proposal brusquely, informing Lord Tywin that he was a good and valuable servant, yet a servant nonetheless.”
Tywin and The Dornish Princess: “What I did not tell you was that my mother waited as long as was decent, and then broached your father about our purpose. Years later, on her deathbed, she told me that Lord Tywin had refused us brusquely. His daughter was meant for Prince Rhaegar, he informed her. And when she asked for Jaime, to espouse Elia, he offered her you instead.”