Mulching will obviously reduce water needs but how much depends on the type of mulch, its thickness, and your climate. Where do you live? watering 2-3 times a day is fine for Arizona but not Florida...

How often to water will depend as much on the soil they are planted in as much as the mulch and how thick that mulch is. If the soil drains very quickly and does not have adequate amounts of organic matter, to help hold moisture, you may need to water daily, or more, but if the soil has adequate organic matter watering may only need be done once a week or so.
Mulches too thin will not help soils retain moisture. I have often seen people plunk down an inch or less of mulch and then complain because then "mulches will not help hold moisture or suppress unwanted plant growth". Mulches need to be 2, 3, inches thick.

I think a good gauge is that once your plants have established and are putting out new growth, see how long it takes you to get to that "wilty" look. That will give you an idea how much much moisture your mulched areas are retaining.

At the same time, you could ignore all that and just water 1-2 times a week, to keep the plants from reaching a stressed and wilty state.

Having said all that, I do notice that my transplanted plants that went to a mulched area need less frequent watering than the similar plants that were not mulched. This just goes to show what the mulch does, which we all know. But it also means that as I observe my plants in the two different areas, I will eventually get an idea of when to water or which area needs urgent water or not as urgent.

Use a left-handed moisture meter.

Here is a link that might be useful: Low Tech Soil Moisture Equipment