I've never bothered with a VPN on Omegle. If you're using text only, there's nothing to worry about. If you're using video, other users can obtain your IP address.

What will other users really do with your IP though? Usually, all they can determine is your internet service provider and general location (e.g., town/city). They won't be getting stuff like your name or address. The other concern is hacking. Technically, a malicious user could do something like launch a denial of service attack against you. Seems pretty unlikely to me that anyone would go through the trouble to harass random Omegle users like that though.

So in my opinion: yes, Omegle is safe to use without a VPN, but a VPN does give you some extra protection. You'll get the assholes who try to scare you by putting your IP or location into chat, but they're just trying to scare you. Just ignore them and hit "next". Don't let these people blackmail you into anything.