While recent Vision-Language (VL) models excel at open-vocabulary tasks, it is unclear how to use them with specific or uncommon concepts. Personalized Text-to-Image Retrieval (TIR) or Generation (TIG) are recently introduced tasks that represent this challenge, where the VL model has to learn a concept from few images and respectively discriminate or generate images of the target concept in arbitrary contexts. We identify the ability to learn new meanings and their compositionality with known ones as two key properties of a personalized system. We show that the available benchmarks offer a limited validation of personalized textual concept learning from images with respect to the above properties and introduce ConCon-Chi as a benchmark for both personalized TIR and TIG, designed to fill this gap.We modelled the new-meaning concepts by crafting chimeric objects and formulating a large, varied set of contexts where we photographed each object. To promote the compositionality assessment of the learned concepts with known contexts, we combined different contexts with the same concept, and vice-versa. We carry out a thorough evaluation of state-of-the-art methods on the resulting dataset. Our study suggests that future work on personalized TIR and TIG methods should focus on the above key properties, and we propose principles and a dataset for their performance assessment. Dataset: https://doi.org/10.48557/QJ1166 and code: https://github.com/hsp-iit/concon-chi_benchmark.