If you happen to select the same home inspector as the other perspective buyer, you can rest assured that they will simply copy over the report and charge you the full amount. It's best to arrange to be there when the inspection happens to prevent this or drive-by inspections.
One of the issues that keeps there from being a neutral third party holding the inspection results is the vast difference in quality in home inspections. Since home inspectors are not liable for anything they miss in the inspections, there are some really poor quality inspectors out there. The third party hiring the inspectors isn't going to be aligned with your interest in hiring the best inspector. Most likely they're going to go with whoever is the cheapest so they can keep as most of the fee you and other buyers are paying them.
In states that license and regulate home inspectors, perhaps part of the requirements could be that every home inspection report has to be part of the public record for the property. If there are multiple inspections, each of them becomes part of the record. A sneaky seller might pose as a buyer and payoff an inspector to give them a good report that becomes part of the record so it would still be in the buyer's interest to hire their own inspector if there is only one inspection report. At some point though, there will be enough inspection reports that your risk of only seeing faulty reports becomes low.
One of the issues that keeps there from being a neutral third party holding the inspection results is the vast difference in quality in home inspections. Since home inspectors are not liable for anything they miss in the inspections, there are some really poor quality inspectors out there. The third party hiring the inspectors isn't going to be aligned with your interest in hiring the best inspector. Most likely they're going to go with whoever is the cheapest so they can keep as most of the fee you and other buyers are paying them.
In states that license and regulate home inspectors, perhaps part of the requirements could be that every home inspection report has to be part of the public record for the property. If there are multiple inspections, each of them becomes part of the record. A sneaky seller might pose as a buyer and payoff an inspector to give them a good report that becomes part of the record so it would still be in the buyer's interest to hire their own inspector if there is only one inspection report. At some point though, there will be enough inspection reports that your risk of only seeing faulty reports becomes low.