I certainly wouldn't log anything as PUT without a signature to say that the training had indeed been carried out.
NOt required and you have no right to ask for any signature for routine training
except for any training flight you specifically wish to use as part of your SEP Class Rating revalidation requirements.
The FI will be PIC on any training flight, unless he/she is flying as a passenger with another pilot operating as PIC. Irrespective of any signature.