There are a very large number of variables, in terms of the internals of the PC at each end, the software used to control the transfer, and the lower level protocols used.
Achieving a reliable transfer also means both ends keeping check of where they got to, and a delay while they agree how far they have got, before they carry on sending. This means there are inevitable overheads to consider, (bits that must be sent back and forth in addition to the data bits of the payload) which consume some of the bandwidth, AND introduce elements of Latency (the time taken for a message to pass from one end to the other, and a confirmation to return to the sender). All of these factors combine, and mean that the data will flow at a rate that is less than the available bandwidth (and often a lot less, when the Bandwidth is large).
Then some software can use compression (recognising chunks that are similar to previous chunks) so that it can send a small reference to the previous chunk, rather than repeating the chunk again).
So the speed at which data can be moved from end to end, is usually a lot less than the Bandwidth (particularly when the Bandwidth is large), but can sometimes be much faster than would be expected (as is increasingly useful when the bandwidth is small).
The choice of Protocol, and the higher level software choices, can make a very significant difference in the speed achieved, but are highly dependent on the specific situation. The unique blend of processing power / memory buffers / Bandwidth / Latency / Numbers and Sizes of files / Compressibility of data that occurs in a particular case.
But to answer the specific question. Given data to be moved between the same devices, over the same network (just running at different speed), the 1Gbps Network is passing bits 10 times faster than a 100Mbps Network and will likely be quicker end to end.
BUT because of the contribution from Overheads, and the speed limitations within the PC at each end, it won't be 10 times faster, and if there are issues with equipment or cabling that cause lots of errors and retries at the higher speed, it is possible that the end to end speed is worse.