There is a general principle that an insurer will cover their responsibilities to the insured and may go after any other party involved (and in effect their insurers if they are insured).
They don't go after their own client outwith the terms of the policy they have with that client.
It's not quite that simple, though. In the case I referred to above, the driver's door of my car was damaged by a cyclist, who didn't have insurance. When I reported the accident to my insurers, they decided that I would need to claim under my own policy, which meant I had to pay the £600 excess. The cyclist reimbursed me for the excess, and I thought that would be that.
But ever since then (Sept 2009), I've been getting letters from my insurers telling me that they are pursing the cyclist for the full cost of the repairs (which was over £3000). They appointed a firm of laywers to recover this money from the cyclist, but this action was launched in my name. That is, the case was lodged with the county court as me (not my insrance company) v. the cyclist. This meant that I ended up dealing directly with the solicitors, and was legally responsible the fees they incurred. Their T&Cs stated that they would recover their fees under the Legal Expenses insurance provided by my motor insurance policy, but also carried the caveat that if their fees exceeded the limit of my cover, I would have to settle the remainder myself. And of course, they could not tell me what the total fees were likely to be - although the hourly rates for the three(!) partners assigned to the case were predictably eye-watering.
Furthermore, the lawyers' T&Cs required my full co-operation in pursuing the case - otherwise they would refer the case back to my insurers and I would automatically become liable for any costs incurred to date. Had that happened, I can only imagine that my insurers would have then taken action against me to recover their loss, on the grounds that I had effectively prevented them from recovering it from the third party. I was also left wondering how much I may have been left out of pocket if I had not had legal expenses cover as part of my policy...
I'm sure this is somehow all part of the gravy train that seems to accompany all insurance claims these days, whereby everyone from lawyers to accident management companies seems to be taking a cut of any payout, but the bottom line was that I was left in a position where I could have been liable for subtantial fees for pursuing a claim in relation to an accident that was no fault of my own in the first place (my car wasn't even moving at time!).