There is no need to use many middleman websites you will find via a search engine. Most of these are *bogus*, even slick-looking ones. More and more banks are offering offshore bank accounts direct. Just get a list of banks in country you're interested in, and go to their web sites. See Google Open Directory here:
http://directory.google.com/Top/Business/Financial_Services/Banking_Services/Banks_and_Institutions/
and here:
http://directory.google.com/Top/Business/Financial_Services/Banking_Services/Banks_and_Institutions/Regional/
and list at EscapeArtist.Com http://www.escapeartist.com/offshore3/banks.htm.
Opening an offshore bank account is like opening one in your high street; meet their criteria, and you're in. The only difference is you're not there in person.
The first thing is to find out whether they will accept citizens or residents of your country. For example, Swiss banks tend not to want US customers; they don't want hassle from IRS.
You will need to prove your identity, and legal existence of your company, if you wish to open an account for it.
If applying by mail, DO NOT PART WITH ORIGINAL DOCUMENTS. Get copies notarised by a notary public. Originals can be used for fraud or identity theft. Or they can get lost.
A Notary Public is a public officer commissioned by State to perform notarial acts. A Notary is an impartial witness. The notary is empowered to issue an apostille.
Apostille - Is a method of certifying a document for use in another country pursuant to 1961 Hague Convention. With this certification by apostille, a document is entitled to recognition in country of intended use, and no certification or legalization by embassy or consulate of foreign country where document is to be used is required.
In practice this means you provide evidence to this man that you are who you say you are, and/or that your company is what you say it is. You take an oath on Bible. That's right, it's not a joke.
Due diligence: Banks need to show they have checked who their customers are, and how they came by their money.
Passport - If you apply by post a notarised copy is needed;
Information about yourself - name, date of birth, address, phone number etc.
Your economic background - documents showing how you earn your money (work contract, bank statement, tax return, company documents);
Origin of your deposits - documents showing how you earned them. If you sell a house, proof of sale, a copy of estate agent's listing, and so on;
Information about your deposits - how much you plan to deposit, and what you plan to do with money once you've banked it.
If opening a company account, you send an apostilled copy of certificate of incorporation to bank providing your account, along with evidence of your identity, an application form, and any other documents they ask for.