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Channel estimation in the presence of frequency offsets is developed for cooperative orthogonal frequency-division multiplexing (OFDM) systems. A two-time-slot cooperative channel estimation protocol is proposed. The source broadcasts the training sequence to the relays and the destination (first time slot), and the relays retransmit the training sequence (second time slot). Pilot designs for amplify-and-forward (AF) and decode-and-forward (DF) relays are derived. These designs eliminate interrelay interference (IRI), which occurs due to the simultaneous relay retransmissions, and minimize the mean square error (MSE). Consequently, the number of AF and DF relays is constrained to be less than lfloorN/(2L - 1)rfloor and lfloorN/Lrfloor, respectively, where N is the total number of subcarriers, L is the channel order, and lfloorarfloor is the maximum integer part of alpha. The pairwise error probability (PEP) of orthogonal space-time coding in cooperative OFDM due to both frequency offset and channel-estimation errors is also evaluated. The optimal power allocation ratio between the source and the relays to minimize the PEP is derived for AF and DF relays. When L < 16, DF relays outperform AF relays in terms of PEP. With L = 4 and 16 active relays, the gap is 9 dB for a frequency offset error variance of 10-3, and this gap increases to about 11.3 dB when the variance increases to 10-2.