Getting chummy perhaps (although I have to pretend not to be a squeeing fangirl on occasion lol). Buying them a drink, most certainly. Asking to be email buddies?* Tbh, that would come across as weird UNLESS you were pretty damn friendly already, or say you were both with the same agent (or had some other business link - in which case you'd use that perhaps). Now, you might swap business cards. But that's still not an invitation to send your stuff/be bestest blurb buddies. Even in that situation, unless I'd known them a while/quite well, I'd never ask them - I'm sure if they want to read my stuff, they'll ask me.just getting chummy and asking to correspond or whatever.
If I really wanted to ask, I'd still go through their agent/publisher.
*I've talked to a few authors at conferences, and seen others initiate convos. Mostly, people are asking them 'So, is Character X going to be in the next book? Why did you kill Y? OMG I love your stuff! There's a plot hole on page 245...' People are mostly asking the author about the author's stuff (or talking about publishing in general). There's nothing wrong with saying 'I'm trying to get published at the moment/will be self-pubbing next week.' But that's a far cry from 'Please make your agent sign me' or 'Read my book and tell everyone how wonderful it is' (Manners would dictate that you might mention your stuff, but you don't push it on others - wait for them to ask if they want to. At least my manners do )
Do people randomly ask authors at conferences to read their stuff? Perhaps. But as with the email, I don't suppose it has a huge positive response rate - in addition to all the reasons stated above an author might not read your work, now you have added 'Huh, so he only talked to me so I'd read/blurb his book.'
Not an impression you want to give, or likely to win friends and influence people.
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