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Yes, the board member can do this, and there would not be an obligation to tell the others present that it is on. While generally speaking Florida is a "two party consent" state when it comes to audio recordings, that only pertains to private conversations. Florida makes it a crime to intercept or record a "wire, oral, or electronic communication" in Florida, unless all parties to the communication consent. See Fla. Stat. ch. 934.03. Florida law makes an exception for in-person communications when the parties do not have a reasonable expectation of privacy in the conversation, such as when they are engaged in conversation in a public place where they might reasonably be overheard.
In other words, it's only illegal when there IS a reasonable expectation of privacy. An open meeting such as an HOA meeting is not a place where there's a reasonable expectation of privacy regarding the conversation, and as such there does not need to be consent regarding the recording. The person recording still should do so out of decency, but there's no legal obligation.
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