Learning to catch callsigns, and write them down, can be difficult at times. Except for when I'm on my club's repeater, in a QSO with other(s) who already know me, I always spell my callsign out phonetically whenever I make an initial call. When I'm on HF I almost always spell my callsign out everytime I ID vs. just saying it in plain language as HF conditions are so variable.
As has been already posted, you can simply ask for the other station to repeat their call and/or upgrade the equipment in your shack to help you hear things better. I have all my ham transceivers and scanners hooked up to their own external speakers. The audio is much improved over the stock in-radio speakers. I also have a Heil mic/boom setup on my Icom 746Pro and the audio reports post-upgrade were well worth the money invested.
Of course equipment alone won't help you to understand the other operator if he doesn't speak clearly into the microphone. For the last 20 years I've been an emergency services dispatcher and it amazes me that the only training that PD/FD/EMS and dispatch personnel get in regards to radio operation is:
"This is the radio. You press this foot pedal and talk. Release the pedal when you're done talking." I've seen dispatchers get so close to the microphone that they almost choke on it and others who sit so far away from the mic that they almost have to yell across the room to be heard. Whether at work or at home I set the microphone up the way I was taught in broadcasting school. The mic is a fist-width away from my mouth at a 45-degree angle. That reduces popped 'P's and other harsh consonants. I also speak a bit slower than I do in normal conversation and use the phonetic alphabet whenever I feel something may not be heard correctly by the units in the field. I get very few complaints, or requests to repeat, from my officers or from those I'm in a QSO with.
It's the same thing for those who operate mobiles and H/T's, ham or professionally. You need to hold the mic at a distance that will allow your transmission to come across clearly. Too close to your mouth and you'll sound muffled or over-modulated. Too far away, or leaving the mic in the holder, and you have to yell across the car to be heard. For me that distance seems to be the same as my desk mics, about a fist-width from my mouth. I also hold the mic at a slight angle from my mouth to avoid the harsh consonants.
Okay...I've rambled on a bit too much here. I know the OP was just talking about logging callsigns and I covered a lot more than that but I'm just trying to help