To achieve this goal, your microphones have to be decent, your recording studio or room should not be made of sound-reflective walls, and you should be wearing (if you are) noise cancelling, or at least noise isolating headphones, to reduce or eliminate microphone-headphone feedback, which further distorts the recording(s).
One podcaster suggested a $100 microphone, the Blue Yeti USB Microphone (Amazon link).
Another podcaster suggested another $100 microphone, the Blue Nessie USB Microphone (Amazon link)
Assuming you are interviewing them remotely, you probably don't need to send expensive microphones to your interviewees to record their part. Audiences are tolerant of some poor sound quality and digital cellular networks and internet connections can often, in conjunction with normal headsets for cell phones can do an adequate job in a pinch.
Some have suggested Lavalier or clip-on microphones, some of which can be had for under $10 on Amazon.Here's an example of a homemade sound shield made of cardboard and foam acoustic tiles:
You should also consider using headphones or a headset of some kind to monitor your audio recording. Most computers can be set up to take input from your microphone and output to your headphones. Some microphones, like the Blue Nessie have a built-in audio out jack for monitoring the recording. Set your output gain/volume so it's slightly louder than the voice you hear in your head, so you'll know if it cuts out and you can investigate the reason.
Wired headphones are probably going to be cheaper. You want the kind with the big headband and over-ear noise isolating cups. You could splurge on noise cancelling, but you want to avoid a situation where your headphones could possibly feed back into the microphone and cause that high pitched feedback screeching. Some have suggested using noise-isolating ear plug microphones. Look for 10 dB or more isolation or noise cancellation. Malcolm has purchased these $16 headphones from Monoprice and they seem to work well (Monoprice link).
Traditionally, audio interview recorders would provide a directional microphone and monitoring headphone headset to each participant, and record each audio track separately for each participant, being able to mix and remix the tracks before publishing them. Monitor headphones would be jacked into a feed mixing all the separate tracks into one channel so everyone could hear everyone else. A good sound engineer would mix those well so the balances would sound good to everyone. Lower budget studios would record the entire interview and only be able to tweak the single recording as a whole.
In that spirit, with one or many microphones, you could potentially do the same thing with audio recording and editing software, including using directional microphones and separate channels for each participant. A free option for software to do that is Audacity.
When asked about software, one podcaster mentioned using Zencastr. Zencastr is a web-based service that records each participant separately and transmits the recordings to the sound engineer using Zencastr. It has a free model, for low output podcasting, which may be good to use to start. Recording parties may need to have always on Internet while recording. More info as we have it.