I have a "community" page on Facebook, which I have used to post comments on other community pages, but they don't show up properly—the only people who can see them are people who already "like" my page. Am I doing something wrong?
That's fairly standard.
While potentially, some people could see your comments if they visited the other community page, it will not show up on their feeds.
Your best bet would be to message the community page and request they post a message to their fans on your behalf. They may be happy to do this depending on what you request.
Failing that you could create a Facebook Ad targeting everyone from that page or group which may gain you some new followers to your page.
Hope this helps ;)
You may need to change the privacy settings of your page to make it public or change the posts public to share it work more audience.
I hope the following link would clarify all possibilities: your-info-on-fb