From my limited experience, oscilloscopes are useful while learning since you can see what is happening in the circuit. For example, you can build an oscillator or filter circuit then have a visual of the results.
In terms of needing one: no, if you can borrow one or have access to someone else's lab. Oscilloscopes have relatively low benefit given the cost. They will either be too low frequency to be of value for many modern applications, or too finicky to learn how to use properly for higher frequency applications (not to mention expensive for higher frequency applications).
In my (also limited) experience I was able to get by with just a high-impedance piezo earpiece I could listen to in order to debug audio-frequency circuits. Ears are surprisingly sensitive detectors and can provide quite a bit of data into your brain. Some higher-frequency circuits you can still debug, but not well or at least not as thoroughly. It's not a perfect solution, but dirt cheap, and consumes a tiny amount of space. It's instant feedback and after a while you can start recognizing (guessing at?) different waveforms and types of signal quirks. I've found that a surprisingly useful part of my kit over the years. Just stay away from high voltage!
The poor-man's scope, yes. I still have an AM-radio-sized battery-operated amplifier from Radio Shack in my toolbox, but it's been a long time since I used it.
An AM radio works too, and works above audio frequencies, wirelessly! The loopstick will pick up harmonics off everything in the circuit so you can hear a lot of what's going on. Helps to know what might be going on though.
I had a boss that had retired the following interview question (maybe it was a PhD qual question, I forget): He'd give the candidate a portable TV and an AM radio and have them measure the horizontal scan frequency of the TV. It was probably a chestnut in some circles.
In terms of needing one: no, if you can borrow one or have access to someone else's lab. Oscilloscopes have relatively low benefit given the cost. They will either be too low frequency to be of value for many modern applications, or too finicky to learn how to use properly for higher frequency applications (not to mention expensive for higher frequency applications).