We get a lot of daft Muslim scare stories in some parts of the press here in the UK, "sharia law" being one of the favourites - this article in the Guardian tries to point out the reality:
On closer inspection, it is clear sharia courts only have jurisdiction on civil matters and everyone must opt in to a sharia court. They only have an advisory capacity and address mainly property and financial matters, and rulings are then only enforceable by civil courts. In many cases, they are understaffed affairs, where one official settles petty disputes and draws up rudimentary documents.
So just like any other kind of alternative dispute resolution - mediation, adjudication, expert-determination, arbitration etc.
Edit - I'm actually over-stating this last bit, "sharia" law isn't nearly as strong as these formal ADR mechanisms.
There are important issues that crop up when the dispute resolution claims a higher authority. The societal pressure to accept such judgements are a lot higher and the arbitrariness and biases (against women, for example) of such judgements are also higher.
The problem with these stories tends to be that some native people in the UK don't really seem to understand the legal system and use these headlines to come to the conclusion that muslims aren't subject to UK criminal law, which is obviously false. Thus the tabloids like to print them because they generate outrage.
As far as I know, they aren't "courts" - they have no legal power. They are just people trying to resolve disputes outside of the courts - which is actually pretty common.
I'm not trying to say that there won't be problems with these kinds of mechanisms, but if UK citizens want to get together and try and work through a dispute based on some religious principles and all parties agree then I don't think we should be trying to stop it.
I think the fear is that having seen the countries these immigrants come from--in many cases, not liberal democracies--the native population is afraid that the immigrants want to make the country more like the old country, and local sharia courts are just the first step.