Now, I haven't actually read through the entire Berne Convention myself...but this page, maintained by Andy Kent, although directed specifically at the anime fan community (and as it was before the digital age), is very helpful in pointing out the important points.
The two points which I then went and found from the Berne Convention are,
Firstly, ARTICLE 8. Authors of literary and artistic works protected by this Convention shall enjoy the exclusive right of making and of authorizing the translation of their works throughout the term of protection of their rights in the original works.
This simply means that translations, scanlations and fansubs are all illegal. Translations are often tolerated for various reasons; it's not worth it for the company to follow up on it, especially if the translator in question is in a different country; for certain languages, there are no commercially available translations; translations are pretty much useless unless you have the original material anyway (applicable to manga mainly) etc.
Scanlations are not legal, not even if you own the material in its original language. To apply it to the manga/comic world, the original company probably won't sue you for it, since you're part of the fan base that's actually paying for it, but a company that has licensed the translated version of the comic won't be getting anything from you.
Fansubs are, needless to say, illegal. See Andy Kent's page for a detailed Q&A about it.
The second point I picked out is more important for worldwide distribution of copyright material. ARTICLE 5(3). Protection in the country of origin is governed by domestic law. However, when the author is not a national of the country of origin of the work for which he is protected under this Convention, he shall enjoy in that country the same rights as national authors.
Any country that has signed the Berne Convention is required to follow it. Article 5(1), which I haven't quoted, states that an author's work is copyrighted in all applicable countries, and this article basically means that an author's work is also copyrighted according to the laws of any country it is distributed in. In other words, any anime distributed in the US is subject to both the Berne Convention's stipulations and US law regarding it's copyright. Simple as that. (As a sidenote, fandom in Japan is much more flexible than in the US, mainly by unstated rule rather than law though.)
Basically, most things which the internet fan community are doing are illegal. Whilst companies might be more tolerant of people who actually purchase their products, it is still not guaranteed that they won't take action if you are caught with a substantial amount of illegal copies, or even worse, making a profit out of these illegal copies. Whilst most people do not partake in the latter activity, we have no right to complain about companies that take steps to cut down on piracy.
As a final note, anyone who lives in a country who has not signed the Berne Convention is only subject to the copyright laws in their own country.