How did dial-up modems make noise when initializing internet connection
Basically all the beep boop noise at the beginning is a bit of low-speed link negotiation, and the really loud white noise PSSSST sound is the actual high speed data going through the line.

Here's a super nice infographic + blog post about it.

http://www.windytan.com/2012/11/the-sound-of-dialup-pictured.html

edit: oops, misunderstood the question.

basically all the phone lines in the house are connected--when one phone is on the line, all the phones basically mirror the sound. 

this is the reason why when you try to use the phone while a modem is under operation, it might kill the link since you disturbed the phone line.

in the real old days, modems literally looked like this:

https://www.youtube.com/watch?v=X9dpXHnJXaE

it was a box with a speaker and a microphone on it that accepted your phone set, and transmitted sounds and received them encoded with FSK.

nowadays there isn't really a reason to physically place a phone set on a box, because the modem itself connects directly to the phone line and does its magic (it's probably better for signal integrity this way).

my memories of using modems don't involve me actually hearing that sound--some modems might have a sound output to reassure that the modem is actually working, or you might have heard it because you accidentally had another phone set connected to the same line.

edit2:

All a modem does in this context is change data (zeros and ones) from your computer to something the phone line can accept (electrical signals that conform to a certain standard, such as voltage and bandwidth).

There is a chip or a few chips that take the bits, use something called a modulation scheme, which all it does is create time-varying voltages that encode the bits. An example is Frequency Shift Keying, which basically means a zero corresponds to a tone at one frequency (let's say 1000Hz) and one corresponds to a tone at a different frequency (let's say 2000Hz). This creates a warble when transmitting continuously (this is the scheme used in that antique modem.)

I believe modern fast modems use something like QAM

As for how these signals are made, it's quite complex. Usually you have something like a DSP processor, which all it does is makes digital representations of continuous analog signals. The DSP modulates digital bits into these digital representations of the analog signals. This then pumps it to a DAC, which turns digital representations of signals into real analog voltages.

So to draw the whole signal flow diagram, you have some sort of connection between the computer, let's say USB, ethernet or even RS-232, and there is probably going to be a processor that takes the PC protocol and extracts the relevant bits that are actually going to be transmitted through the phone line. This processor passes it to a dedicated DSP processor, which does the aforementioned business and passes it to a DAC, and you might have some analog signal conditioning/buffering circuitry and out goes the signal through your phone line.

As for the reception, there is an ADC which is the inverse of a DAC, and the same DSP demodulates the analog voltage back into bits, which goes back into the computer.