There are a lot of tools to do this. If the website has any advertising, then you can probably get the statistics using something like Google Ad Planner, or whatever ad network that they use (they disclose that because people who are looking to buy advertising want to know how many page views they get).
If the website doesn’t have any advertising (this is the case in your question), then there are tools that can estimate the amount of traffic a website gets. They do this by tracking the webpages that a percentage of internet users visit, and then assume that the rest of the internet visits the same pages. Again, this at best provides a rough estimate of a website’s traffic.
Two websites that provide traffic estimates are Alexa and Quantcast, though there are probably more out there. I would recommend taking data from multiple services, so that way you have less of a chance of getting inaccurate data.
Another tool is Google Trends, which gets its data from the amount of search traffic a website receives. It doesn’t show the actual amount of traffic a website receives, but it does show whether there was a change in traffic that the website receives.
You can also guess how much traffic a website receives by using many factors, like what keywords the site ranks high in, or how many comments the site gets (if they have comments), or based on how many visitors similar sites receive. This isn’t very accurate, but it may be the only option available to you.
As you can see, there are different options with different strengths and weaknesses. Accurate, digital, legal competitor surveillance technology is seldom free. Even if you’re not buying the data from the competitor (who would never sell it to you! Plus it might be fiddled…) – you’re still paying someone. Either someone who resells ISP network data or someone who has ‘done something clever’ to get more accurate Google-traffic forecasting than the rest.