Letting your partner drive has obvious advantages (shorter rides, no chance to do damage). But are there any negative aspects to it?
Do you for example still get the distress calls during missions?
Yes, you "miss" distress calls if your partner drives as this is pretty much a "fast travel" option. You will also miss out on finding landmarks of importance.
I have no idea if this is made up for later on in the game if you decide to ignore all of them on a case by case level (as in if you choose to miss 1 distress call does it get moved to the next case?).
Yes, you still get distress calls during missions. Think of it this way, when you are on a case, no matter what part of that case you are up to, you are still in the mission. When I last played the cases seemed to be "spaced out" in the sense that you could only get 1-2 distress calls per case. It could have been my luck though!
Using the trip skip option (ie, letting your partner drive) will make you miss any radio calls to respond to street crimes. You will also not get credit for any landmarks passed or any other items of interest along the way. However, this does not cause you to miss anything permanently.
There's an option after you finish an entire desk (usually 4-6 cases) to go into "Streets of LA" mode on that chapter. If you choose this option, you'll be put with your partner for that desk, and you'll have the option of driving wherever you like. The distress calls will show up on your map, and you can choose to respond to any of them you like. There's no way to miss these, and you can't permanently fail them. There is one set of street crimes per desk, and they appear randomly during cases on that desk. This can sometimes get annoying if the dispatcher is constantly sending you halfway across the map during a case...
The landmarks are available in any of the investigations as well as in the Streets of LA mode in any desk. There are also hidden items and secret garages with unique cars in them. The Rockstar Social Club website has a map that contains the locations of all of these items, and it tracks which ones you've already found. If you're going for 100%, it's a vital resource!
I would suggest using the trip skip option whenever possible during the cases, as driving the car yourself can lead to accidents and property damage, which counts against you in the final case rating. You can always take a break and do street crimes later if you so choose.