If you're looking for alternative expressions to use for the commonly used term "press statement," there are many options available. Depending on the context or the audience you may want to use one of the following synonyms: a declaration, official statement, announcement, notice, communication, or message.
Alternatively, you could describe the communication for a specific purpose such as a public relations release, media briefing, or a press release. In each case, the intention is to communicate information for public knowledge. The wording used in each format may vary, but the essence of the message should be consistent. By expanding your language options, you can better tailor your message for optimal impact and reach.