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The short answer is yes, many politician do write their own speeches, but this isn't necessarily a good thing for their audiences. Professional speech writing is a skill, and not every legislator or public official has the inherent language skills to do it well. Politicians may be far better at other job duties, such as drafting new legislation or making informed voting decisions. Because an effective speech can take hours or even weeks to prepare, it is not unusual for senior politicians not to write their own speeches.
A politician may be called upon to speak to a wide variety of constituents on a wide variety of subjects. It would be impractical to assume any public official would have specific facts and figures on all those subjects stored in their minds. Professional speechwriters or staffers with strong language or research skills are often recruited to write specific speeches tailored towards specific audiences. Politicians may tweak the speeches for clarity or time constraints before delivery, but few modern political figures have the time to write their own speeches.
This has not always been the case, however. Political figures during the 18th and 19th centuries were generally expected to be strong orators and statesmen, which meant they had the innate ability to write their own speeches. President Abraham Lincoln, for example, was widely admired for his speech writing and oratory skills, even for an address delivered at the Gettysburg battle site which lasted less than two minutes. Presidents were also expected to write their own State of the Union speeches and other major addresses to the public or Congress.
Modern politicians, on the other hand, are often expected to be strong business leaders or skilled legal experts, not necessarily strong speakers or statesmen. The task of drafting a memorable speech often falls on professional writers with a background in public relations or advertising. This is primarily to prevent misunderstandings or even international incidents triggered by a poor choice of words. Politicians are always free to participate in the speech writing process, but their ideas may have to be skillfully modified by a professional writing staff in order to create a strong speech peppered with memorable sound bites and heightened language.