The general-purpose cache-coherent many-core server processors are usually designed with a per-core private cache hierarchy and a large shared multi-banked last-level cache (LLC). The round-trip latency and the volume of traffic through the on-die interconnect between the per-core private cache hierarchy and the shared LLC banks can be significantly large. As a result, optimized private caching is important in such architectures. Traditionally, the private cache hierarchy in these processors treats the private and the shared blocks equally. We, however, observe that elimination of all non-compulsory noncoherence core cache misses to a small subset of shared code and data blocks can save a large fraction of the core requests to the LLC indicating large potential for reducing the interconnect traffic in such architectures. We architect a specialized exclusive per-core private L2 cache which serves as a victim cache for the per-core private L1 cache. The proposed victim cache selectively captures a subset of the L1 cache victims. Our best selective victim caching proposal is driven by an online partitioning of the L1 cache victims based on two distinct features, namely, an estimate of sharing degree and an indirect simple estimate of reuse distance. Our proposal learns the collective reuse probability of the blocks in each partition on-the-fly and decides the victim caching candidates based on these probability estimates. Detailed simulation results on a 128-core system running a selected set of multi-threaded commercial and scientific computing applications show that our best victim cache design proposal at 64 KB capacity, on average, saves 44.1% core cache miss requests sent to the LLC and 10.6% execution cycles compared to a baseline system that has no private L2 cache. In contrast, a traditional 128 KB non-inclusive LRU L2 cache saves 42.2% core cache misses sent to the LLC compared to the same baseline while performing slightly worse than the proposed 64 KB victim cache. In summary, our proposal outperforms the traditional design and enjoys lower interconnect traffic while halving the space investment for the per-core private L2 cache. Further, the savings in core cache misses achieved due to introduction of the proposed victim cache are observed to be only 8% less than an optimal victim cache design at 32 KB and 64 KB capacity points.