It becomes quite dangerous when they achieve the same rights as individuals. In my opinion, the most acute danger from this course of action is that businesses can achieve a much greater hold over government and government policy. This allows the cycle to continue where business gets more powerful and actual individuals are victimized by it. Reasonable limits on business are necessary.
If we took a more jaded view of business interests and their reach into the social order, there might be several concerns when companies are considered the same as citizens. At this point, the company and the individual are entitled to the same treatment. The problem here would be that the company can exert power and influence that far outstrips one individual. There has to be some level of check to limit the business encroachment into politics and social orders. History has shown that business interests can demonstrate a strong collusion with government, to influence the direction of policy and legislation, and to be able to secure the same rights between both individuals and companies seems to offer some level of empowerment to an institutional order that is already quite empowered. I think that there needs to be a stronger sense of government distinction between these forces and their influence to petition the government or the maximum use of any other rights that citizens have because the company possess greater instruments of power than the singular citizen.
It seems to me that the most relevant concern (especially recently) is that calling a corporation a legal person gives it all the same rights as a human being.
The reason that this has come up recently is that it is the basis for the Supreme Court decision that just said that the government cannot regulate political speech by companies. It said that companies have the right to spend their money on political speech just like regular individuals do. Many people think that this will give companies too much power to affect (or simply buy) our political leaders.
Joint stock companies are treated as persons created by law to enable them to be able to be able to take some actions legally valid otherwise valid only for individuals, and similarly be subject to restrictions and liability of some legal provisions, of law. Treatment of companies as persons created by law does not really give them the status of citizens. For examples, companies cannot vote in elections as citizens do. Similarly no country in the world confers to companies available to humans.
I am not aware of the the supreme court judgement cited in the post above. The judgement appears to be about the right of companies to spend money on political speeches rather than freedom of speech. Perhaps what is being implied that a company is justified in spending money in pursuing its valid commercial activities, and making political speech could be a part of such commercial activities.
It is worthwhile to note that the joint stock companies, unlike common citizens are not allowed to engage in any business they like. The companies can engage in only such businesses and activities as are permitted under the article of association of the company.
Finally it should be remembered that companies are created and allowed to function to help human beings that own them. Companies are not in competition with humans. If there are any isolated incidents of legal provisions giving powers to companies hurting the interests of people, I am sure, once these are known, people and governments will take actions to correct such anomalies.