Yes they can.
If profiles are enabled for your meeting, then participants must enter a first or last name (or both).
A participant can edit their profile at any time during the meeting but all their previous as well as their future messaging and polling contributions will be associated to the new profile.
However, as a meeting host you can change the profile settings of your meeting and you can do this on the fly. For example, you may wish to collect some anonymous feedback at the end of your presentation. However, if you want to retain any identified polls or messages, remember to download your meeting report from the Data menu in your meeting before you change the profile settings.