I have two users on SE. Both of them are registered in SO, and one of them is tied to other SE communities accounts like Arqade, ELL, etc. How can I sign into SO with one account while signing into other communities with another account? When I log out from one account from one community it seems I'm log out from all communities.
Keen's answer is the best way to go if you didn't create the two accounts to specifically have two accounts.
If you did want two accounts, for instance to keep your work and personal usages of the sites separate, then merging your accounts isn't what you're looking for. So long as there is zero interaction between your accounts (for instance, voting) and you're not using the accounts to do things you shouldn't be able to (for instance, voting on a post twice) then there really isn't a rule against two accounts. It's discouraged but not against the rules.
Keeping that in mind, you cannot accomplish this with on-site means unless you want your accounts merged. There are a couple options for off-site means, however:
Use two different browsers
This method is probably one of the easiest. If you sign into one account on Chrome and one account on Firefox, for instance, then you never have to sign out of either account to use the other. It's not perfect, but it works well.
Use two different browser profiles
Very similar to the above approach, but in one single browser. Some browsers, for instance Chrome (and I believe Firefox), allow you to have the same browser running with different sessions with them. The cookies and all are different between the two sessions. This would give you the same effect as running two browsers, but with only one browser to work with.
This would also be achievable with Incognito/Private browsing, as mentioned in the comments. Using Incognito/Private browsing might be preferable if you only log into the one account occasionally but the other far more often. Log into the account you use most on a regular session of your browser, and the other in Incognito mode.
Use two different devices
This method is, of course, not available easily to everyone. Not everyone has two computers or a computer and a smart phone, or a computer at all. But if you do, you could always set up one device with a certain account, and put the other account on a different device. It's not nearly as handy as the other two methods, in my opinion, and it does require having two devices you can access Stack Exchange on, but it is an option.
At the bottom of every page on Stack Overflow, there's a 'contact us' link. Click it. In the 'What can we help you with?' dropdown, select the merge profiles option. Then provide links to both profiles, and submit your request.
A Stack Overflow employee will then be able to help clean up your extra account. Once this is done, signing in on one site should then sign you in across the network.