If you clear the tracking cookies in your web browsers, you usually have to log back in to websites.
Sometimes your web browsers may clear the cookies themselves it seems when a certain alloted memory space is taken up. Here again, you will have to log back in to previously visited websites.
I use Firefox and to a lesser extent, Internet Explorer........and both of these browsers will let you save your passwords in them if you want. Even if you have to log back-in, the browser remembers your screenname and password if you saved it previously.
I bet the other browsers do the same too.
If you are concerned about saving your screenname or passwords on a computer that others may use, then you should give portable apps a try. You can put just about all the browsers on a USB flashdrive (along with many other free apps)......and then take the flashdrive itself from computer to computer. This way, you leave no trace on whatever host computer you use, and keep all your web browsing personalizations in the tip of your hand. And it's all free.