Say I have a remote data center in LA and my office is in San Diego. When you say I have a site to site dedicated line between the data center and my office, does it mean fibre lines which do not connect to the internet but just between the office and the data center? How is this even possible? Do they lay the lines just for this purpose? Do they dig up the path between the cities just for this request? I cant visualize. Please explain
Today this would likely be done by laying a fibre from your office back to a nearby PoP and from the DC to a PoP near the DC. Then most providers would create a L2 VPN (such as an MPLS pseudowire) between the PoPs across the service providers network between PoPs. It wouldn’t actually be a dedicated fibre from end-to-end.
The two main exceptions are if the two sites are within close proximity of each other (in relative terms, say 20Km) the sites might both hair-pin through the same provider PoP. Or if you purchase what is usually called “Dark Fibre” then you are typically paying for dedicated end to end fibre strand(s).
When providers lay fiber they don't just lay one fiber. Over land and on short underwater hops they lay a large bundle because fiber is cheap. Transoceanic fibers need integrated optical amplifiers which reduces the practical fiber count. Your long distance link will run mostly over existing infrastructure with possiblly a small ammount of new fiber for the final connections.
There are various levels at which you can buy into the fiber infrastructure depending on your bandwidth/latency/jitter requirements.
You can buy a dedicated fiber path, this is often known as "dark fiber" because the customer typically supplies the equipment to "light" the fiber. This is most common on shorter links where the cost of the fiber is relatively small and so the costs of WDM or electronic muxing are not justified. Those with very high bandwidth needs may also buy dark fiber to run their own WDM system over.
You can buy a wavelength in a WDM system. This is a common option for high bandwdith links over long distances. The provider will generally operate the specialist WDM transcivers and if needed optical amplifiers and use a non-wdm fiber for the final link to the customer.
You can buy a line that is muxed electronically. This is common for relatively slow (sub 10Gbps) links. There are various systems for doing this including ATM, PDH, SONET, MPLS etc. Some systems only allow fixed bandwidths while others allow for a burstable service. The final link to the customer may be either copper or fiber depending on bandwidth requirements and what is available locally.
Short answer: "Back in the day" maybe they did this. Very quickly, you just got connected to the nearest POP (point-of-presence) with IP over ATM over copper, from which you got allocated a VC (virtual circuit) over a mutualized ATM line. You got a "T1", a "T3"... Between 1995 and 2005 inter-POP ATM lines were progressively replaced by IP over MPLS over copper then fiber, and last mile was somewhat replaced by ATM over DSL over copper and now by IP over fiber.
It does happen. We've had the luck to get a few dark fiber pairs spliced off when a larger trunk was laid down along a major road just three years ago. In fact, one of our locations doesn't have any local servers any more. It's just some 2 km and you can't see a difference between a local and a remote server.