They can be detected. You just have to measure in 90% H2O, which is common in protein analysis. You will then have a big solvent peak, that you can (partially) suppress with different methods. If the water peak is not in the way for your other peaks this allows you to observe OH and NH peaks in your sample.
If the water peak is just a bit in the way you can shift it by changing the temperature. Higher temperatures shift the water peak to lower ppm, lower temperatures will shift it to higher ppm. Do mind the effect of temperature on your analyte, and with water as solvent, don't go too low, I think that going below 10 degrees centigrate could result in artifacts, but I have to admit, I have never tried.
To answer your second question "So any functional groups that exchange with D2O to get H to D transfer cannot be detected in NMR. Am i right ?"
If you have exchangeable H like COOH, OH, NH and you run 1H-NMR in protic solvent like D2O, CD3OD then you will not detect it easily since it will exchange with D of the solvent ! so you are right in that.
However, I have checked the solubility of poly(acrylic acid) and found that it is soluble in THF, [http://www.polymersource.com/Resources/Solubility.pdf] which is aprotic solvent. So, you may run your 1H-NMR in THF-d8 [http://www2.chem.umd.edu/nmr/reference/isotope_solvent.pdf] and you will detect your COOH proton.
Indeed, the detection of protons that have fast exchange with the solvent D2O is complicated as Adel said above. Thus your acid proton will not appear in the acid region but shows up as HOD (increasing the water peak, which might be present in a small extent in your D2O). What I'd like to add that not only H/D exchange but also H/H exchange can occur. Thus if you record your sample in H2O (unlocked or by using solvent suppression with a capillary of a deuterated solvent) you will get also a single peak (slightly downfield-shifted to that of water alone) because the exchange of these protons is way too fast on the NMR time-scale. This exchange will also occur when you use wet non-protic solvent, i.e. you will get an average chemical shift between your acid and water peak. That is why it is important to use dry solvent for your goal. If you don't want to use and expensive solvent for the detection of your acid group, you can stay with D2O and record HSQC or HMQC. These are easy to record 2D, 13C-1H NMR correlation spectra, which are going to show you a cross peak between your 13C(OOD(H) ~ 171 ppm and 1H HOD ~4.7 ppm. A single 13C NMR is also very characteristic for the acid group (it might take somewhat longer to record than a HMQC).
Additionally, if you run the spectrum at 85-90 ºC and PH= 1 you would improve very much the resolution of your spectrum, showing the signals due to methylenes (2.0-1.6 ppm) split due to syndiotactic (one broad peak at 1.8 ppm) and isotactic (two multiplets at 2.0 and 1.6 ppm) dyads that could you allow to estimate the tacticity of the polymer.