As far as I know, Skype and SIP accounts are not compatible. So, no you won't need a Skype account to talk via SIP. However, you will need a SIP account. By the same token person at the other end will need a SIP account too, not to mention a working and properly configured SIP client.
Unlike Skype, I found configuring SIP a bit more difficult, particularly because there are multiple SIP service providers and settings may be a bit different for each.
Both Skype and SIP allow calling regular phones and receiving calls from phones. Unfortunately these are not free in either for Skype or for SIP service providers.
If you just want to talk and already have a gmail account, you can try Hangout. It works without any download/install, (tested with Chrome browser). You can talk to people with Android phones and tablets as well as PCs, using WiFi (or wired Internet for PCs). Skype apps are available for phones as well.
In any case, you will need an account somewhere. The account is what identifies you to the other Skype/SIP/Gmail users. If you don't have an account there is no easy way for someone to call you (and you to call someone else without an account).
One alternative is to use Viber. Not that it does not need an account, but it uses your (mobile) phone number as your account number. So anyone who knows your phone number can contact you. See How to install Viber in Ubuntu? for more on Viber. I have never used it, so I don't know much about it.
Hope this helps