I would say that to have the actual formal job title of 'Statistician' or 'Data Scientist', usually a Master's or above is required. To perform that type of work, it is not entirely necessary.
In fact, I have found that people from all sorts of backgrounds perform statistical analyses in a work setting. But indeed many jobs that primarily use statistics every day do usually require a Masters or above (pharmaceutical experimental design, quantitative finance, etc.). A lot of it depends on the job and the network you build to get it but also because they may require more in-depth understanding of concepts from a graduate level.