The AC volts setting on the digital VOM is designed to measure the rms value of the sinewave. Inexpensive meters will only be accurate for sine waves, and only for lower frequencies. 1kHz should be low enough to read accurately, but that will depend on the meter.
The rms value is the "equivalent" DC voltage that would produce the same average power as the AC signal.
Better digital VOMs (DMMs) will read accurately for higher frequencies (but still rarely above audio frequencies), and will also read accurately for non-sinusoidally signals. In other words, they give True RMS readings.
All that being said, the scope shows you the actual waveform. You're seeing the full peak-peak voltage of the sine wave. To convert between an RMS value to a peak to peak value, you would multiply the RMS value by 2.828. Thus, 1Vrms = 2.828Vpp. So, what you're seeing is normal.