I'm a US citizen living abroad with a dual-citizen son. I can tell you that there are advantages and disadvantages. The benefits are that they can travel freely to the US and live and work in the US without having to obtain a green card. Also they can travel on either of their pasports largely depending on which country they travel to and they will qualify to receive assistance from any US consulate or embassy when overseas. They will be legally entitled to vote in any federal elections in the US when they turn 18, although if they have never actually lived in the US in practice they can't because no state will allow them to register to vote in that particular state.
The disadvantages are that when they start working they will always have to file a tax return in the US, regardless of where they actually live. For the most part they will receive an exemption for US taxes for any income they receive while working overseas with the exception of self-employment income, if they are legally self-employed then they will have to pay self-employment tax in the US in addition to any tax they pay overseas (some, but not all foreign countries have a self-employment double-tax agreement with the US, though which mitigates this). For me to avoid this tax I had to form a foreign corporation and work for that corporation so I'm not legally self-employed.
Another disadvantage is that they will be required to register for the US selective service when they turn 18 (the draft). There has not actually been a draft since the Vietnam war, though, so this is not likely to become an issue, but it is certainly something to consider.
As stated by the parent they can always renounce citizenship later and avoid the tax and selective service issues, but this is expensive (about $2500USD).
Also speaking of expense, having to file two tax returns means additional accountants fees and additional paperwork, especially if the country you live in has a different tax year than the US (which is very common). Having to maintain two passports is another extra cost as well, but not very expensive when you spread the fees out over the life of the passport.
All of the above said, I made an informed decision to register my own son as a US citizen and I do agree that the benefits outweigh the down sides, but it's certainly not a "nothing to loose" situation, there are downsides and it pays to make an informed decision with full knowledge of them.