The insurer pays 0-X, then the reinsurer (and their reinsurers respectively) pay X onwards.
But, in insurance companies buying insurance, there's some differences.
I'm not licensed but I do know the "Quota Share" structure is quite industry accepted for a few different lines (property being one, casualty, auto). So the % of risk retained versus ceded at various points to reinsurers is exactly the contract structure. Within that structure there are some limits and retentions that get hammered out during the contract and negotations ($XXX,XXX per occurrence of X, Y, Z, etc). But the point of this is that yes a company can retain XX% and reinsurers, usually several, split up dibs on the other XX%, sometimes in layers as well.
It's pretty interesting stuff and the numbers of what's out in the markets for underwriting is definitely an eye opener.
For example if the probability of fire is 1%, but due to their limited geographical focus they were covering an entire building of 30 apartment and their coverage is 100 dollars well their rate should be 1 dollar + x%, but if their is a fire their payout will be 3000 dollars due to that correlation of all the apartments burning down. In that case the reinsurance company would have to cover that. If I were Berkshire I would have to price the policy higher for the increase risk.