I worked for Astrill so I can tell you the claims attributed to them on that page are mostly bullshit.
1) We do not keep logs
Reality) False. They do keep logs. With reguard to http traffic, they log every single single page loaded for every customer using their VPN servers. They log your account number, the requested url, the time of the request. This information is kept for months at a time and then thrown into a journal system.
2) We do not disclose your customer information
Reality) Mostly true. Astrill is mostly concerned with closing the accounts of people who abuse their network. If anyone asks they will say they can't identify what customers are doing, but in reality is Astrill is constantly scanning through their logs looking for people running scripts which hit google/facebook (or any other site) too frequently, and then those accounts are disabled.
3) We may enable logs on a server in case of network abuse
Reality) False. Http requests are logged on all servers and that data is kept for a minimum of three months. Always. When I asked about this discrepancy between what they say publicly and what they do privately I was told we would do whatever it takes to run the company (even if that includes deceiving the customers).
4) We do not store any files, we provide IP transit only. Therefore, we can’t fulfill any "take down(s)"
Reality) False. Astrill receives a whole bunch of take down notices every single day and then blocks users from reaching those destinations. One of the uses of the logs they keep is to locate copyright troll IPs. We developed scripts to compare take down requests (which include our offending torrent IP addresses and time stamps) to locate the trolls, and then block our clients from connecting to them. We still did a lot of take downs, but by locating copyright troll IPs (by running scripts on our logs) we were able to block many IPs.
5) The majority of users use shared IP space therefore we are not able to identify the customer
Reality) False. We knew exactly who registered what accounts and logs every one's activity. We can and do identify your account. We close accounts each day for abuse. We were working on a system to throw abusive accounts onto one server. I don't know what progress was made in this endeavor.
6) P2P activity is allowed on many of our servers
Reality) True. These are the servers marked with a star in the connection tools we provide.