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"Home water distillers get rid of dangerous chemicals"
My experience in alcohol distillation leads me to doubt that they get rid of all of the volatile chemicals, as some of them would be carried over in the first part of the distillation and be concentrated in the first liquid produced.  To get rid of them, the first liquid (called the heads) from the still would have to be discarded.

The same concentration happens with less volatile chemicals in the last liquid from a distillation (called the tails), so that should be discarded also.

Any nasty volatile chemicals that stills do remove will now be wafting about in the atmosphere of your home for you to breath anyway, so maybe the still should be only run outside.

Some units include a charcoal filter to remove these chemicals, but you can use a charcoal filter without a still and without removing the minerals.

"Look at all the nasty crud in the bottom of the still, I used to drink that!!!"
That "nasty crud" is minerals.  I have also suspected that consuming distilled (or de-ionised) water would strip minerals from your body to replace the minerals removed by the distillation, increasing the risk of osteoporosis, osteoarthritis etc.  You may have learned a bit about osmosis when you were at school, but we tend to forget these things when confronted with well presented advertisements.  My reasoning is,  when you drink mineral free water, you don't pee mineral free water, therefor, somewhere along the line, you have been robbed.

I now find that I am not the only one with this opinion.  See Early Death Comes From Drinking Distilled Water by Zoltan P. Rona MD MSc