This is partly redundant with the answer of Victor Protsak, but I want to emphasize his point 3. The main reason I am very happy that my library has many paper subscriptions to math journals is that every week I have a dozen issues to browse. When the title of an article rings a bell, I take a look at the abstract and, sometimes, it convinces me to read the introduction or at least the main theorems. For some journals, e.g. Annals of Math or Acta, I read all abstract regardless of the subject. It is not uncommon that I cannot understand a word, but still I cross nice or important results in a large portion of the mathematical subjects.

This is clearly not sufficient to adress your question, though: I had not heard of the works of Okounkov or Werner before they get their Fields medals.

This is partly redundant with the answer of Victor Protsak, but I want to emphasize his point 3. The main reason I am very happy that my library has many paper subscriptions to math journals is that every week I have a dozen issues to browse. When the title of an article rings a bell, I take a look at the abstract and, sometimes, it convinces me to read the introduction or at least the main theorems. For some journals, e.g. Annals of Math or Acta, I read all abstract regardless of the subject. It is not uncommon that I cannot understand a word, but still I cross nice or important results in a large portion of the mathematical subjects.

This is clearly not sufficient to address your question, though: I had not heard of the works of Okounkov or Werner before they get their Fields medals.