When a group of people gather to form or work for a corporate entity, the group itself (ie the corporation) does not have the same constitutional rights as the individuals who make it up. For example, it has no right against self-incrimination. Other rights are transformed into duties: it does not have a right to an attorney such as the state would provide one, for instance, but instead must be represented by an attorney at its expense.
For all that, it keeps some rights. Famously, the Supreme Court held some time ago that corporations, as such, enjoyed free-speech rights. And since the right to speak entails the right to pay others to speak for you, they had the right to purchase political advertising. This continues to be a source of controversy.
The next question making its way up through the courts is whether corporations have freedom-of-religion rights. And that question arises, sideway, via Obamacare. Since the new health law mandates coverage for birth control pills, companies owned by religious organizations that oppose family planning claim that the mandate infringes on their religious freedom. And so we come full circle. Companies, in and of themselves, have no right against self-incrimination; they do have free-speech rights. Do they freedom-of-religion rights? Only time and the Supreme Court will (eventually) tell.
PS – There are other facets to the corporate personhood debate, but most are statutory rather than constitutional. For example, companies may not resist freedom-of-information requests because the information would be private and embarrassing; they have no statutory privacy rights under FOIA. But they are persons enough to be part of a civil lawsuit in tort or contract, as we as a society has long accepted. The list goes on and on.