Below is the situation about the two pronunciations as I know in Mainland China.
In modern Chinese dictionaries, you'll find the pronunciation for 说服 is "shuo fu", which means persuade.
However, the single character 说 normally should be pronounced as "shui" when it means to persuade or more specifically, to make people agree with your opinion. For example 游说(you shui, basically means to lobby).
Here you can see the source of the confusion. That is since 说服 means to persuade, and when it means to persuade, why 说 isn't pronounced as "shui"?
The only reasonable explanation I can come up with, although seems a little bit retrospective, is that 说服 as a whole word means to make people agree though talking.
I'm no linguistic, but you can see 说服 as a verb-adverb phrase, in which 说 is the verb, 服 is an adverb stating the result of the action. In this sense, 说 literally means speaking, and 服 means affirmative.
So you can think it as 说(shui) equals 说服(shuofu) in their core meanings. However, nowadays, 说(shui) is often used in a more politically aggressive context and only used in "游说"
In day-to-day use
In everyday, scenarios, most people I know would use 说(shuo)服, but 说(shui)服 is also acceptable and nobody would confuse them.
Personally, I would pronounce it as 说(shuo)服, in sentences like:
I persuaded him to come to the party.
But in certain situations, I would go against the dictionary and say:
I will persuade the board to kick him out!
To me, there are three scnarios that I would use 说(shui)服
the subject matter is important to me and I really want to win the case.
I'd like to make the other party agree with me no matter what, even it means that I would have to distort facts or being misleading or deceitful in the argument.
Or in a politically aggressive context like the example sentence above.