You definitely do not have to go through your current employer to get training. There are many organizations that provide Lean Six Sigma training including GoLeanSixSigma.com. Courses typically range in depth, but it’s always best to go through a workshop with examples to practically apply the tools can concepts.
Although it is not necessary to go through your employer to access training, one of the benefits of getting the training through an employer is that they may reimburse you for costs or pay for your training. There are many public courses offered online and live. Costs vary by provider and location.