I want to record our meetings, especially discussions about projects, because I often miss some important details they tell me and my writing cannot catch-up. However, I don't feel like I need to tell them as they may feel it's an invasion of privacy. I have no malicious intent whatsoever, just really so that I can review the project details / comments and not miss out any.
"However, I don't feel like I need to tell them as they may feel it's an invasion of privacy"
That right there is a big reason not to do this. You already feel that some people wouldn't want it done, and your immediate reaction is to simply not tell them rather than take their concerns on board.
Another consideration to take on board that it may simply not be legal in some jurisdictions - under UK law, you can record a private conversation without the consent of all parties, but you break the law if you play that recording back to another party without consent from all the recorded parties.
Similarly, US law differs state to state - some states are "one-party", and some states are "all-party" (and Massachusetts bans secret recordings outright), so you may be breaking the law by recording without consent.
Never ever do things this like this in secret. It will come out sooner or later and will be looked upon badly, to the point where you may be fired.
As Alexander commented under your question:
It is the responsibility of the person managing the meeting to make sure you understood everything that was discussed. It should also be their responsibility to produce a documentation of all important points of the discussion
Go to your manager or to the person managing the meeting and discuss the issue with him. Not as a complaint, but ask him for guidance, so that you can work this out together.
In that conversation you could even bring up the suggestion of recording. If you decide to do so, the recording device should be clearly visible and switched on/off to all participants, you would have to make clear to everyone what the purpose of the recording is, and you should also destroy it a.s.a.p.*
In addition, one thing surprises me: if you are supposed to take notes (either for yourself or for others), why don't you just say things like:
- Sorry, can you repeat that. Did you mean X?
- So, to make it clear for me, did you mean we have to Y?
That would be taking responsibility instead of complaining afterwards.
Also, from experience, what really helps: go over your notes again (or work them out better) directly after the meeting (first thing). At that moment things are still fresh in your head, and you may even be able to contact a meeting attendee for clarification (it is still fresh in their head as well).
* You have to take away the fear that the recording can be used to hold people to their words three weeks later - nobody likes that.
However, I don't feel like I need to tell them as they may feel it's an invasion of privacy.
You are wrong!
You do need to tell them, particularly if they would feel that it's an invasion of their privacy. It's simply the right thing to do.
I have no malicious intent whatsoever, just really so that I can review the project details / comments and not miss out any.
Then explain this to them - that you have no malicious intent and you just want to avoid missing out on important details.
They may consent. Or they may decide to have someone take notes for the group during meetings and distribute them afterwards - resolving your details problem. Many companies appoint a "scribe" for important meetings, sometimes on a rotating basis.
Either way, go along with their decision.
Recording without the permission of others is not only unethical, but in many places illegal.
If you "DO" record meetings, the recorder should be in plain sight, and you will need to remind people that they are being recorded. If someone says something potentially embarrassing, offer to delete immediately. The recording should then be transcribed and destroyed ASAP.