I've always wondered how other countries and rulers came to know about the new world. Columbus and Spain knew they were on to something big, even if they didn't know exactly what. Why didn't they keep it quiet? Why spread the knowledge rather than take everything for themselves?
His discovery spread through the press, of course!
J.H. Elliot writes in his book The Old World and the New that "Columbus's first letter was printed and published nine times in 1493 and had reached some twenty editions by 1500." (pg. 8-9) Suffice to say, it was initially pretty popular. Though, notably, it was thought to have a small audience that didn't reach the public, and instead consisted of "humanist and religious circles." (pg. 12) [1]
The discovery would have been impossible to keep quiet. After all, to simply arrive in Castile the voyage needed to pass through the Portuguese port - and was, in fact, taken captive by the Portuguese upon his initial return. What a welcome, right?
The only reference I can think of to secrecy would be the sugar growing techniques of Portuguese in Brazil, which were a closely guarded secret until the land was seized by the Dutch West India Company. [2]
Of course secrecy existed, but rarely officially. Traders smuggled things as traders tend to do. I don't have a reference, but I remember reading that angry natives and blacks would, in the Indies, sell secrets to European privateers consisting of when the Castilian treasure fleets would be leaving, and when they would be crossing the isthmus.
As to your last point, Spain and Portugal had the new world for themselves for a long time. Remember, it was expensive to send fleets, and only the extremely wealthy nations could afford to do so. Poorer countries wanted a piece of the pie, too, but without the initial investment they had to resort to privateering missions to take a bite. The French largely stuck to Northern America and the English had no real ability to send official fleets until the early seventeenth century. Even then, their expeditions largely relied on private investment to get going.
Hope I could be of some help! Sorry the paragraphs are so discombobulating!
[1] J.H. Elliot, The Old World and the New, 1970, pg. 8-9, 12
[2] J.H. Elliot, Empires of the Atlantic World, 2006, pg. 105
although i'm not terribly familiar with Columbus, specifically, i do know that a collection of his letters, as well as other European explorers, made their way back to Europe and spread like wildfire. it was exciting to find a whole chunk of land where there was none before (well, none known, i mean).
check out De Orbe Novo for a compilation of the letters concerning Central and South America. i presume that similar volumes exist for the rest of North America, but i don't know them by name.
edit: you can find either the first or second (can't recall right now) volume for free download through Google Books.