I wrote to The Vegan Society for reliable information on this subject, and they told me:
Probiotic cultures are made in a number of different ways – the best thing to do is to contact the manufacturer and ask them how it is produced. Many of them are produced using dairy as a source, so these would not be suitable for vegans.
When I was in China I bought soya milk from the supermarket, and I was unable to read the labels, so I asked a Chinese friend to check the ingredients for me, and he said that some of the soya milks contained dairy.
I noticed a few years ago that M&S were selling a soya milk that contained vitamin D3, presumably from lanolin. I believe the product was labelled "dairy free" and "suitable for vegetarians" but I assume it was not vegan.
These products are not necessarily manufactured with vegans in mind, so we cannot assume that any added probiotics they contain are vegan. To be sure, you would need to ask the manufacturer about their process, as I was advised.
If the product is labelled vegan, you can be more confident, but you still need to trust whoever is applying that label. Personally, I tend to accept that a product is vegan if it says it is, but I have heard a few stories of mis-labelled items. The Vegan Society Trademark is a strong guarantee:
Any product which has our Vegan Trademark has been confirmed to contain no animal ingredients, to not have been manufactured using animal ingredients and not to be tested on animals.
Here's the full list of standards for the Vegan Society Trademark