I would say that to have the actual formal job title of 'Statistician' or 'Data Scientist', usually a Master's or above is required. But to actually perform that type of work with a different title, it is not entirely necessary.
In fact, I have found that people from all sorts of backgrounds perform statistical analyses in a work setting (although not necessarily in a correct manner). But indeed many jobs that primarily use statistics every day do usually require a Masters or above (pharmaceutical, quantitative finance, economics, etc.). A lot of it depends on the job and the network you build to get the job. If you can convince someone that you are good enough for the job, then you may be able bypass the educational requirements in the job description. But also keep in mind that some of the reasons that they stress Masters or above (at least based on my experience) is the level of Mathematics and independent research that would be required to work independently in the given position.