The question of how legally effective it is might be a secondary one; if it is practically effective in discouraging people from extending credit to your estranged spouse/child or whatever in the expectation that you will cheerfully pay up, then it’s doing its job.
also was used in correlation with filing for bankruptcy.
The practise is starting to sound something like those disclaimers that businesses sometimes put up “no refunds … not responsible for defective merchandise” and so on and so forth. Some of them also don’t have legal standing, in that companies have been known to claim they’re not liable for stuff the law clearly says they ARE liable for. But they still do it, just in case it works.
Probably if you’d seen other people put “not liable for debts” in newspaper notices, or if some friend advised you that this was how it was done, you’d do it simply because it couldn’t hurt. And sticking a notice in the newspaper is probably cheaper than actually going to a lawyer to discover whether it did any good in the first place.