If I work in a law firm and have not signed any employment contract, am I legally allowed to leave the firm and work for a counterparty?

It's not clear to me whether you want to work for the party opposing your former firm's client, or for the law firm representing that party.  Either way you can work for them, but you can't divulge any privileged or confidential information you learned during your employ with the first law firm.  Even assuming that you have never agreed to maintain the confidentiality of the matters you worked on for the law firm Iit doesn't have to be in writing), and that, as an assitant, you are not subject to any ethical statute or rule, the confidential and privileged information to which you refer might well be considered trade secrets, which a former employee in any business may not use.  Although I am not aware, off the top of my head, of any case on point, I believe that any Court is likely to hold that allowing you to disclose attorney/client privileged information to your new employer would violate public policy, and would prevent you from doing so and/or prevent the recipient from using the information  If your new employer tried to use that information to their advantage in the lawsuit, the Court would prevent them and possibly sanction them.  If your new employer was a law firm, they might be subject to disciplinary action as well.  If you do go to another law firm, they would be wise not to allow you to work on that matter.