For a basic everyday user both these proxies are the same since they both provide anonymous browsing. However an individual with knowledge about the technicalities of web will know that there are some basic differences between them. However despite the difference you can buy private shared proxy and you can even buy some cheap private shared proxy.
There is a difference in speed
If we are to look at a shared proxy, there are multiple users using it at the same time. This results in an overloaded bandwidth which then results in an increasingly slow internet speed. The fact these proxies are free attracts many users which is why their speed tends to be quite slow. On the other hand this is not the case with private proxies. This is in use by one person specifically. So with a private proxy not only is the speed higher but the risk of sharing your private web whereabouts with anyone are completely eradicated.
There is a difference in security
Once again, shared proxies are used by many people often at the same time. This has a risk of your personal information being exposed to hackers or spammers. Access to passwords can be gained through shared proxies. However, the private proxies are completely secure and provide individual cyber security like no other. The reason being that they are used by one user only.
There is a difference in anonymity
The shared proxies are often run by sites where there is a risk of your data being exposed. They are sometimes even transparent and might be passing on your IP address to the website. A private proxy on the other hand as an almost absolute chance of hiding your IP address. This means your data and web whereabouts are completely concealed and secure.
There is a difference in reliability
The reliability factor is extremely low with shared proxies. You never know that you might as well be using a compromised proxy server which is being overviewed by a hacker. Free proxies are always a risk and one can never be completely sue about them. You might not even see a particular shared proxy after a few uses because they come and go and are not reliable at all. It goes without saying that these risks do not exist with a private proxy since they are under use by one specific user.
There is a difference in advertising and spam
With using shared proxies you have a risk of your email addresses being sold off to companies. Other than that since these proxies are coming from unreliable sources they might leave you with some unwanted on click advertisements and banners. This effects the malware and your device. The private proxies do the opposite. They protect your IP address from being attacked by any unwanted software which might be harmful for your device.
To sum it up, the private shared proxy might be free but is not as secure as private proxy.