Explain and you might get more useful replies
Private does tend to be more efficient as they don't have the financial backing like public services they're in it to make money they don't make money they go bankrupt. Efficient in the way of cost to service ratio maybe not actual quality of service. If public services run out of money they have many more options than a private entity.
There's a counter-argument, namely that private has inherently higher de facto costs for providing a service because, as you rightly point out, "they're in it to make money". Their profit is therefore added to the cost of actually providing the service (but not counted as a cost, despite the fact that it obviously is). So in order to be more efficient in terms of cost alone, even disregarding quality entirely, they must reduce costs (in comparison with public sector) by more than the profit they take.
Then there's hiding costs by moving them back to the public sector. Privatisation of public services provides one way - private businesses using only the parts they can profit from, leaving the public to pay the cost of the rest of it. The benefits system provides another - the business can cut its costs by reducing wages, leaving the public to pay for benefits to bring the employees' incomes up to the cost of living.
I think the correct answer to the OP's question is "it depends". On various things - what area of service/business you're looking at, how the organisation is run, whether it's selling a product or providing a service, whether you count all costs or just the ones which support one side or the other, etc.