Are Researchers at Private Companies Allowed to Publish Papers?
Speaking as a researcher at a private for-profit company, who also publishes frequently, it all depends on the nature of the business model and the position.

Most people at companies do not publish, either because the company has no interest in publishing or because their particular job doesn't involve much of scientific interest. For companies that are involved in cutting edge work, however, scientific publishing can gain credibility and visibility for the organization just like it does for people at universities. This is the case for the company I work for, particularly given that many of our projects are federally funded research where the government is paying us in part to disseminate knowledge.

Nor is there any necessary conflict between protecting intellectual property and publishing. The three main strategies for protecting IP are copyright, patent, and trade secret. Of those, only trade secret is incompatible with publishing, and it is generally the most fragile and least used in any case.

The biggest obstacle to publishing in private industry is simply priority and time. Writing a manuscript is a lot of work, and if management doesn't see much benefit, they will likely want you to use those hours for things more directly connected to the bottom line. You might be able to publish out of hours, but that can be a more complex negotiation.

Bottom line: it really depends on the particular circumstances. The best way to know if you will be able to publish is to look at whether others in the group you are joining publish. If so, you probably can. If not, there are likely major obstacles, whether formal or merely pragmatic.