Since the suicide bombings earlier in the year in Riyadh, police have been setting up more checkpoints around Jeddah.
It makes the traffic flow a lot slower but it's safer for everyone.
They don't say anything about Westerners travelling together (i.e. unmarried men and women), but you have to be VERY careful if a single western woman is travelling by herself with an arab man.
Just make sure you have your iqama/passport with you when out and about.
jeddah is quite a big city with an estimated population of 3,000,000 people. however, there's no proper system of transportation; one of the main complaints visitors often make about jeddah. traffic is getting more congested and very busy. a 10-min journey may take up to 30 minutes. since the petrol is cheap and car rental is very affordable, everyone has got a car!
there's only one bus route between North and South Jeddah, via Medina Road, but it's only for men and it's not quite handy. it's really difficult to get around the city without a car, as you cannot walk. streets are quite big and wide for pedestrian.
watch out for the traffic jam during the holy month of Ramadan. it's just beyond your imagination. all those who never leave their home get to leave it in ramadan! as the working day is shorter, starts late and finishes early, so all people leave almost at the same time, so you end up with a massive traffic jam. instead of spending a 15-minute journey, you find yourself caught in a 1-hour crammed traffic.
this season is like christmas. everyone hates to go to work. everyone wants to do shopping and dine out. everyone does lots of family visits and religious occasions. it's very very very busy!
the best bet to avoid this problem is to wake up very early so to drive in peace, and then head back for home later than people leave. yes, you end up having a longer day than everyone else, but it's your call!