Check your Website Speed
Speed is more important than you may think...
The speed of your website has influence on how your users experience and use the site. Nobody likes slow websites, and users often leave these kind of sites. And, search engines like Google rank fast websites higher than slow ones. This will become even more important in the future...
Quote from Google: "Through both internal studies and industry research, users show they prefer sites with a great page experience. In recent years, Search has added a variety of user experience criteria, such as how quickly pages load and mobile-friendliness, as factors for ranking results." see this page.
To be able to judge the speed of your website objectively you can use a few tools:
Google PageSpeed Insights
A simple page, but it can give you important information about the performance of your website. You may see problems that smaller images alone don't solve, but the results tell you if your images are used with optimal compression or not. See: https://developers.google.com/speed/pagespeed/insights/
This website gives an enormous load of details about your webpage. Speed, Security and so on. You can test the speed of your webpage from various locations in the world. See: https://www.webpagetest.org/
Tiny PNG Analyzer
This page let you analyse the images on your website, and gives you an idea of improvements that can be made. See: https://tinypng.com/analyzer
For owners and buyers of websites
Don't let the folks you are buying a website from tell you that it's not that important. Speed is important. But it is not always easy te make websites fast. Loaded Wordpress sites, and that is loaded with plugins and templates, are very difficult to optimize for speed. Especially on cheap hosting... But you should trust the outcome of the test websites linked above. And you should demand from website builders they get green results from those tests. Really.
Even if the "Tiny Image Processor" and tinify.com are not able to put your website in the acceptable speed bracket, at least it will have improved your speed. You can try plugins for your CMS, a faster server, the use of a CDN or something like that....