Please understand I can only answer the first question in any detail.

In the classical period, there really were no scientists per se. Only well off people with an interest in nature and the natural way things occurred. Their science was based primarily on observation and to put it bluntly, "thinking about things". There was little experiment but there was debate, believed to take place at symposiums, which in those days was essentially a banquet or party. Points were usually backed up with observations taken in situ, rather than experimentation to eliminate potential other causes for behaviours. Mathematics was also discussed in similar ways, but proof was presumably far more commonly produced due to the nature of the discipline. In this era, Aristotle laid out a way of looking at the world, revolving around (to simplify it), cause and effect. He reasoned that there must be a "why" to each effect and it can be argued that this is the seed from which the scientific method grew. He becomes more important later.

The Academy of Gondishapur is an interesting aside which I personally feel belongs here. Emperor Khosrau I (who by the way, was later known as the philosopher king and was a really, really good ruler), gave refuge to many philosophers and Christians who could speak Sirriac. He got these people translating ancient Greek and Sirriac texts, which is one of the reasons why the Caliphate had them to copy out later. He also invited many Indian and Chinese scholars there, to speak on things like astronomy and mathematics and importantly, write this stuff down. A Nestorian Christian church where all this took place, soon became a renowned center for learning and effectively became the equivalent of a university.

The fall of the Western Roman Empire, left western Europe in a bit of a slump scientifically as everything was a bit unstable for several centuries. Weak nation states lead to stagnation and those with both the means and the will to investigate the world were few and far between, meaning that there were few scholars to share their ideas with and even fewer opportunities to network.

In the east though, science flourished. With the aforementioned Academy of Gondishapur and later, the House of Wisdom in Baghdad, the east had specific locations where scholars would meet and work, with vast libraries and most importantly, a focus on Aristotle's way of viewing the world and even experimental procedure, with the "Canon of Medicine" by Avicenna actually using medical trials to back up it's claims. The book was used well into the 18th century. In short, they met up, corresponded, debated and wrote books.

Books like these, propagated across Europe in the 8th and 9th centuries, where the thinking was mostly done by clergymen, seeking to understand the nature of God's creation, at least until the founding of the first universities, such as Bologna in 1088. From here, it's understood that alchemy (the fore runner to chemistry) became more and more important. The Aristotelian approach and the alchemical approach of science being experiment and result based meant that science gradually grew away from dialectic reasoning in favour of experiment. Scientists then published their proofs in books if possible and frequently corresponded with contemporaries. In short, the sharing of ideas was mostly done in self published books or by letters, though where possible, they undoubtedly directly discussed with their colleagues. Books were rarer though, due to the great expense involved.

I apologise for the over abundance of context of the overall state of scientific development, however, I thought it necessary for you to understand not just the methods by which ideas were exchanged, but also the establishment of centres for learning and methods by which those ideas were being developed at the time.