Most of the time, I would do 2 on one side (RHS), then 2 on the other side (LHS). If you have a healthy threshold for mouth opening and want to just get it sorted in one visit, it is possible.
Is it safe to be done under local anesthesia? Yes.
Removing 4 teeth at one go and related considerations:
Pre removal - If the 4 wisdom teeth are fully out and non surgical extractions, then by all means have 4 out at one go under local anesthesia should be a relatively quick and straightforward procedure.
During surgery - your jaw gets really tired half way through! When you have a wisdom tooth taken out, bite props can be provided to aid mouth opening so that the surgeon has a clear visual field, but this still requires your mouth to be open for a significant amount of time.
Post removal - If you need a surgical extraction to remove impacted wisdom teeth (ie they are not growing straight) then having all 4 out would mean that you have to be prepared for a soft diet and facial swelling / bruising ( happens sometimes, not all the time) for 1 - 2 weeks. This is why I tend to advise 2 appointments so that at least you have 1 good side to eat with while the other side is healing.
Other factors - General rule of thumb is that younger patients tend to heal faster. Make sure you are healthy pre surgery. Not advisable to do wisdom teeth extractions when you are under the weather. For comfort of all 4 wisdom teeth to be removed, consider conscious sedation which can easily be arranged. Usual down time is up to 1 week, depends on how many teeth are taken out.
But having said that, everyone is different so it really depends on the individual!
All the best and feel free to inquire further should you have more questions.