First: this is entirely normal, and not something to be concerned about whatsoever. Children have to learn social behavior, and one of those behaviors is to work out their problems without hitting. Most four year olds hit other children from time to time; they will learn over time that there are other ways to solve their problems, and stop, if you take the right approach.
Don't try to compare your method of discipline/etc. to the daycare's; it's okay, and normal, that they're different. It's not your job to deal with the things that occur at the daycare - it's their job.
For the most part, you probably shouldn't do anything directly about the individual reports, either. Unless you witness the incident, the time delay is just too much even at four - an hour or two later, the child won't connect whatever you do with what happened. If you ask them what happened, they almost certainly won't be able to tell you in any meaningful way - because they don't know, for the most part. They're still learning why they do things! So don't try to focus on the individual incidents - try to focus on teaching them how to do better. You could consider a "good day reward", perhaps - four years old is old enough to at least have some idea of that, and it can reinforce the positive days. (At two, for example, even that isn't really effective - they're too young - but at four, it should have some impact.)
Your job is to teach your child how to handle their difficult emotions, and how to resolve conflicts, without resorting to violence. Talk to them about their feelings. Roleplay situations that might come up. If the daycare says they hit another child due to taking a toy away, roleplay that situation. Give them toe tools to handle the situation, and give them the background to understand their emotions and to put words, names, to their emotions.
Take a look at this article on Positive Discipline; you can't really do the top half of their suggestions (which are focused on "at the time") but the bottom half are great examples of how to talk about processing their emotions and handling conflict, like this:
Teach children that feelings are different from actions. Feelings are never bad. They are just feelings. What we feel is always okay. What we do is not always okay.
Help children brainstorm ways to deal with feelings that are respectful to themselves and others. One possibility is to tell people what you don't like. Another possibility is to leave the scene if you are being treated disrespectfully.
From VeryWellFamily, some suggestions about how to give names to their feelings:
For many young children, a lack of appropriate vocabulary is the number one reason they hit. Whether a playmate has yanked a toy out of their hands or mommy isn't giving them the snack they want, rather than try to find the right words to express how they feel, it is much easier to simply use their hands.
(a bit later:)
If preschoolers can express themselves verbally, they may be less likely to physically act out. So start building their vocabulary. When they get mad at their brother for taking a book out of their hands, teach them to say, "I am mad!" When they are upset because they have to stop playing and take a nap, it's OK for them to express, "I am sad!"
Finally, work with your daycare to find out what triggers are causing these outbursts. Your child is not a psychopath, just randomly hitting people; they're doing it because of something they can't handle. Find out what those are, and then use that in your roleplay. At the same time, talk to the daycare to see what they can do to avoid this. Is it one or two children? Then keep your child away from them. Is your child getting too "amped up" and then losing control after? Ask the daycare to help them cool off when they're too wired. There's a lot they can do to help you, and if they're a good day care they should be happy to talk about it.
