Answer: Essentially, when a lateral joins a new firm there is a concern that the potential information disclosure by the lateral breaches the duty of confidentiality regarding certain clients. Firms want to make sure that no conflict of interest exists that could expose them to potential liability if they hire the new attorney. Firms take this very seriously, which is why when they get "serious" about a lateral candidate they begin making endeavors to ascertain that no conflict of interest exists by running "conflicts checks" on the candidate's current and previous clients.
To answer your question, under no circumstances should you ever give notice to your current firm prior to clearing conflicts. While it is rare, occasionally candidates do not clear conflicts at the proposed firm and they cannot join due to a matter that they worked on for a client. You are at the most risk of not clearing conflicts if your practice area is Intellectual Property or is Litigation-related. These areas tend to be a lot more contentious. We rarely see transactional candidates have any kind of conflict that prevents them from joining their new firm, but they should still be cautious. There's nothing worse than giving notice and then not having a new job, so it's always better to be safe and wait.